Document 97628

IEEE TRANSACTIONS ON PATTERN ANALYSIS
4
AND MACHINE INTELLIGENCE,
VOL. 22,
NO. 1,
JANUARY 2000
Statistical Pattern Recognition: A Review
Ani1 K. Jain, Fellow, I€€€, Robert P.W. Duin, and Jianchang Mao, Senior Member, /E€€
Abstract-The primary goal of pattern recognition is supewised or unsupervised classification. Among the various frameworks in
which pattern recognltion has been traditionally formulated, the statistical approach has been most intensively studied and used in
practice. More recently, neural network techniques and methods imported from statistical learning thsory have bean receiving
increasing attention. The design of a recognition system requires carefut attention to the following issues: definition of pattern classes,
sensing environment, pattern representation, feature extraction and selection, cluster analysis, classifier design and learning, selection
of training and test samples, and performance evaluation. In spite of almost 50 years 01 research and development in this field, the
general problem of recognizing complex patterns with arbitrary orientation, location, and scale remains unsolved. New and emerging
applications. such as data mining, web searching, retrieval of multimedia data, face recognition, and cursive handwriting recognition,
require robust and efflclent pattern recognition techniques. The objective of this review paper is to summarize and compare some of
the well-known methods used in various stages of a pattern recognition system and identify research topics and applications which are
at the forefront of this exciting and challenging field.
Index Terms-Statistical pattern recognition, classification, clustering, feature extraction, fsaturo selection, error estimation, classifier
combination. neural networks.
+
1
~NTRODUCTION
B
the time they are five years old, most children can
recognize digits and letters. Small charactcrs, largc
characters, handwritten, machine printed, or rotated-all
are easily rccogiiizcd by the young. The characters map be
written on a cluttered background, on crumpled paper [ir
inay even be partially occluded. We take this ability fnr
grantcd until wc facc the task of teaching a machine how to
do the same. Pattern recngnition is thc study of haw
machines can ohserve thhc environment, learn to distinguish
patterns of intcrcst from their background, and make sound
and reasonable decisions about the catcgories of the
patterns. Jn spite of almost 50 years of research, design of
a general purpusc machinc pattcrii recognizer remains an
elusive goal.
The best pattern rccognizors in most instances are
humans, yr?t we do not understand how humans recognize
patterns. Ross [140] emphasizes the work of Nnbcl Laureate
IJerbert Simon whosc central finding was that pattern
rccognition is critical in most liitman decision making tasks:
"The more relevant patterns at your disposal, the better
your decisions will be. This is hopeful news to proponents
of artificial intelligencu, sincc computers can surely be
taught to rccogiike patterns. Indeed, successful. computer
programs that help banks score credit applicants, help
doctors diagnose disease and help pilots land airplanes
Y
depend in some way on pattern recognition ... We need to
pay inuch morc explicit attciition to teaching pattern
recognition." Our goal here is to introduce pattern recognition as the best pnssible way of utilizing available sensors,
processors, and domain knowledge to make decisions
automa tically.
1-1 What is Pattern R e e q r M o n ?
Automatic (machino) rccognition, duscription, clnssificrltion, and grouping of patterns are important problems in a
variety of engineering and scientific disciplines such RS
biology, psychology, medicine, marketing, computer vision,
artificial intclligcnce, a i d remote sensing. But what is a
pattern? Watanabe [1633 defines a pattern "as oppnsitc of a
chaos; it is an entity, vaguely defined, that could be given a
iiamo." For cxamnplc, a pattem could be R fingerprint image,
a handwrilten cursive word, a human face, ui' a speech
signal, Given a pattern, its recognition/classi~icatioi~
may
consist of one of the following two tasks [163]: 1) supervised
classification (e.g., discriminant analysis) in which the input
pattern is identified as a member of a predefined. class,
2) unsupcrvisad classification (e.g., clustering) in which the
pattern i s assigned to a hitherto unknown class. Notc that
the recognition problem here is being posed ns a classification or catcgoriza tion task, whore the classes are either
defined by the system designer (in supervised classification) or are leamed based on the similarity of patterns (in
unsuperviscd classiFication).
Interest in the area [if pattern recognition has been
renewcd rcccnkly duc k o cmcrging applications which are
not only challenging but also computationally more
demanding (see Table I). These applications include data
mining (identifying a "pattern," e.g., correlation, or an
outlier in millions of multidiincnsioiial pattcriis), document
classification (efficiently searching text documents), financial forecasting, organization and retrieval of multimedia
databascs, and biometrics (personal identification based 011
5
JAlN ET AL.: STATISTICAL PATTERN RECOGNITION: A REVIEW
TABLE 1
Examples of Pattern Recognition Applications
various physical attributes such as h c c aiid fingerprints).
Picard [I251 .has identificd n novcl application of pattern
rccognitioii, called affective computing which will give a
computer the ability to recnpiizc aiid cxpress emotions, to
respond intelligcntly tr) human emotion, and to employ
mcchanisms of cinotion that contribute to ratinrial dccisioii
making. A cotnmnn charactcristic of a number of these
applications is thaL the available fcatmes (typically, in the
thrnisands) arc not usually suggested by domain cxpcrts,
but must be extracted and optimized by data-driven
procedures.
'['he rapidly growing and available computing power,
whilc enabling faster processing nf hug" d a h suts, has also
facilitated the use OT elaborate and divcrsc methods for data
analysis and classification. At the same time, demands on
automatic pattern recognition syslcms are rising enorinotisly dire to the availability of largc databases and
stringent pcrformancc rcquiremeizts (speed, accuracy, mid
cost). In many of the emerging applications, it is clcar that
110 single approach for classification is "optimal" and that
multiple methods and apprriachcs have to be used.
Cunscqucntly, combining several sensing modalikics and
classifiers is now a comtnoiily uscd prrlcticc in pattern
recognition.
Thc dcsign of R pattern recognition system csscntially
involvcs tlic following three aspects: I) data acquisition aiid
pi.eprocessing, 2) data representation, mid 3) decision
making. The problem domain dictatcs thc choice of
seiisoi:(s), preprocessing tediniquc, ruprcsentation scheme,
and khc dccision making model. Tt is genwdly agrccd tlint R
well-defined and sufficiently colisbrained recognition problem (small intraclass variations and large inlerclass
varialinnsj will luad to R crimpact pattern rcpreseiitiition
and a simple docision making stratcgy. Lcnriiing from a set
nf oxnmplcs (training set) is an important and desired
atrributc of most pattcrn rccognition systems. The four best
known approaches for pattern recognition are: 1) templak
matching, 2 ) statistical classification, 3 ) syntactic or structural matching, and 4) neural networks. These modcls arc
not neccssarily independent and sometimes the same
pattern recognition method exists with different interprctntioiis. Attempts have been made to design hybrid systciiis
involving multiple models [57].A brief description and
comparison of these apprnachcs is given bclow and
summarized iin Table 2.
1.2 Template Matching
( h e nf the siinplcsl and earliest approaches to pattern
rccognition is based on template matching. Matching is R
gcncric operation in pattern recognition which Is used t o
dctcrniinc tlw similarity between two entities ( p i n t s ,
curves, or shapes) of the same type. Tn template matching,
R template (typically, il 2n shape) or a prototypu of the
pattern to be recognized is available. The p a t k m to be
recognized is matched against the stored template while
taking into accoiiiit all allowable posc (translation and
rotation) aizd scale changes. 'I'hc similarity measurc, often n
coi:reliition, may be nptimizod bascd on the available
training sct. Often, the template itself is learned from the
training sct. Template matching i s computationally dcmending, but the availability of faster processors has now
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 1 , JANUARY
2000
TABLE 2
Pattern Recognition Models
-7IEI
[
I
Itwogriitinri h d i i m
__
..~-
~
Corw1aI:iou r l i s l nricr
~
.5
+
i i i r n w i'c
Diswiriimaiit fuiictivri
-.
I---
.
Nrvt.iwk hict,iim
made this approach more feasible. The rigid template
matching mcntioncd above, whilc cffcctive in some
applicnlion domains, 1x1s a number of disadvantages. For
instance, it would fail if the patterns are distortcd duc to the
imaging prticcss, viewpoint change, or large intraclass
variations among tlie pat terns. Deformable template models
[69] or ruibbcr slicct defornnations [9] can be uscd to match
patterns when thc dcfurmatioii cannot be easily explained
or mndclcd directly.
1.3 Statistical Approach
In thc statistical approach, each paitcrn is represented in
terms of ri Ccaturcs or ineasui:ements and is viewed H S a
point in a d-dimensional space. Tho goal is to choose those
features that allow pattern vectors belonging to different
categoi'ies to occupy compact and disjoint regions in a
rl-dimcnsionnl feature space. 'l'hc effcctiveness of the
reprcscntntioii space (feature sct) is determined by hnw
wcll patteriis fi4om different classcs can bc separated, Given
R set of training patterns from each class, the objective is to
establish decision boundaries in the feature space which
separate patterns bclonging to different classes. In the
statistical decision theoretic approach, the decision boundaries are deteriniiicd by the probability distribu~ionsof tlie
patterns belonging to each class, which musk cithcr be
specified or learned [41], [a].
One can also takc a discriminant analysis-bascd apprtmch to classification: First a paramclric form of tlie
decision boundary (e.g., linear or quadratic) is specified;
then the "best" decision boundary o f Lhc specified form is
found based on the classification of tminiiig patterns. Such
bouudarics cart be constructed using, for example, a iiiean
squared error criterion. '['hc direct boundary coizstruction
approaches are supported by Vaynik's philosophy [162]: "If
you pr)sscss a rcstricted amount of informahn for solving
Liome problem, try to solve the problem directly and never
sulve a more general problem as a n intermediate step. It is
possible that the available information is sufficient for a
direct solution but is insufficicnb for solving a more general
intermediate problem."
C l ~ ~ s i i i m r iwriw
o~i
1 Classiiiixriciii wriic
h
p
liiilcn, gminin:.w
t ----
Typical Clitcliuii
A<!cr!II1.nrlc~WImr
h.Ic:tu squari: error
I .4 Syntactic Approach
'in many rccognition problems involving coniplcx patterns,
it is inorc appropriate to adopt a hierarctiical pcrspective
where a pattorn is viewed as being composed of simple
subpattcrns which ace themselves built from yct simpler
subpattcrns [56], [121]. The simplest/clementary subpatterns to bc rccognizcd are called prinliliucs and thc given
complex pattern is represented jn terms of the interrclationships bctwccn these primitives. In syntactic pattcrn recognition, a formal analogy is drawn butwccn tlie structure of
patterns a n d the syntax o f a lnnguagc. The patterns arc
viewed as sentences bclonging to a language, primitives arc
viewed as the alphabet cif the language, and the sentences
are generated according to a grammar. Thus, a largc
collecLiun of complex patterns can be described by a small
number of primitives and grilmmatical rules. 'lhc grainmar
for each pattern class inust be inferred from the available
traiuing smiplcs.
Structural pattern recognition is intuitively appealing
bccausu, in addition to classification, this approach also
provides a description of how the given pattern is
constructed from the primitives. This paradigm has been
used in situatinns whcrc: the patterns have a definite
structure which can bc captured in terms of a set of rulcs,
such as EKC wavchrms, textured iiiiages, and shapc
analysis of contours [5h].The implementation of a syntactic
approach, huwcvcr, lcads to many d.ifficu1tics which
primarily h a w to do with the segmentatimi of noisy
patterns (to detect the primitives) and the inference of the
grainiliar from training data, V u [56] inlroduccd the notion
cif attributed grammars whicli unilics syntactic and statistical pattern recognition. The syntnckic approach niay yield
a combinatorial explosion of possibilities to be investigated,
dcmanding large training seis and very largc computational
efforts [I 221.
1.5 Neural Networks
Neural networks can bc! viewed as massively parallel
computing systems consisting of a n extrcmcly large
numbcr of simple processors with inally inkrconnections.
Neural network models aftcinpt to iisc some orgauizatioiial principles (sticli a s learning, gcncralimti.on, adaptivity, fault tolerance and distributcd rcprcsentation, and
7
JAlN ET AL.: STATISTICAL PATTERN RECOGNITION: A REVIEW
computation) in a network of weighted dircctcd graphs
in which thc nodos arc artificial neurons and directed
edges {with weights) are connections botwccn netiron
outpiits and ncuron iiiputs. The main characteristics of
neural networks are that they have the ability to learn
complex nonlinear input-output relationships, USCL SPquentitial training prucedures, and adapt thcmsclves to
tlic data.
The most commonly used family of neural networks f o r
pattern classification tasks [#3]is Lhc fwd-forward network,
which inclitcles inultihyer perceptron and Radial-Basis
Function (RBI?) 11c tworks. Thcsc networks are organized
into layers and have unidirectional connections belweeii the
layers. Anothcr popular network is the Self-Organizing
Map {SOMJ, or Kohonen-Network [92], which is mainly
uscd for d a h clustcring and fcahirc mapping. The learning
process involves updating network architccturc and coilncctim rvciglits so that a network can efficiently perform a
specific classification/ clustering task. 'l'lie increasing popiilarity of ncurnl network models to solve pattertl recognition
problcms has bccn primarily due to their seemingly low
dependence on domain-specific knowledge (relative to
mndcl-bawd and rulc-based approaches) and due io the
availability of efficient learning algorithms for practitioners
to
USC.
Neural networks provide a new suite of nonlinear
algorithms Tor h t u r c cxtmction (using hidden layers)
and classification (e.g.,multilayer perceptroiis). In add ition,
cxisling fcalurc cxtracticin and classification algorithms can
also be mapped on netirill network architectures for
efficient (hardware) implcmcntaLinn. In spitc of thhc seeiningly different underlying principles, most of thc wellknown neural network models are implicitly equivalent or
similar to classical statistical pattern recognition methods
(scc Tablc 3). Riplcy 11361 and Anderson et al. [ 5 ] also
discuss this relationship between neural networks and
statistical pattern recognition. Anderson et al. point out that
"ncural nctworks arc! statistics for amateurs... Most NNs
conceal the statistics from the user." Dcspitu ~ ~ L ' SsiinilaC
ritics, neiirel iictworks do offer several advantages such as,
unified approaches for feature extraction and classification
and flexible proccdurcs for finding good, moderately
nonlinear solutions.
1.6 Scope and Organizatlon
In the remainder of this paper we will primarily review
statistical mcthods for pattern representation and classification, cmphasizing recent dcvclopmunts. Wlwnever appropriate, we will also discuss closely rclatcd algorithms froin
the neural networks literature. We omit the whole body of
li ternturc on fLizzy classification and ftizxy clustering which
a r e in our npiniot-t beyoiid the scopc o f this yapcr.
Intercskcd rcadors can rcfcr to thc wcll-written books on
fuzzy pattern recognition by Hezdek [I51 and [IC,].In most
of tlic st'ctions, tho various approaches and methods are
summarized in tables as a n easy ancl quick refercnce for the
reader. Due to space constraints, w e are not able to provide
ninny dctails and wc' hnvc to omit some of the approaches
and the associated references. Our goal is to emphasize
those approaches which have been extensively evaluakcd
and demonstrated to bc useful in praclical applicntioiis,
along with the new trends ancl ideas.
'I'he literature on pattern recognition is vast and
scattered in numerous joitmals in s c r x x ~ l discipliiies
(c.g., applied statistics, machiiic learning, neural nctworks, ancl signal and image processing). A quick scan of
the table of contents of all the issues of the K E E
Tmnsactions 011 Pattern Annlysis m d Machine IrifclliK~nce,
aincc i k first publication in January 1979, reveals that
approximately 350 papers deal with pattcrn recognition.
Approximately 300 of these papers covered ttie statistical
approach a n d can be broadly categorizcd into the
following subtopics: ciirse of dimensionality (1.5), dimensionality reduction (501, classifier design (175), classifier
ctmbinntion (lo), error cstimatiun (25) ancl urlsupcrviscd
classification (50). In addition to the cxccllcnt textbooks
by Duda and Hark [44]," Fukunagn [58], Devijvur and
Ktttlei: [3Yl, Devroye et al. [4:11, Bishop [18], Ripley [:I371,
Schurmann [1471, and McLachlnii [105], we should also
point out two excellent survey papers written by Nagy
[ill] in 1968 and by Kanal [89] in 1974. Nagy described
the early roots of pattern recognition, which at that timc
was shared with researchers in artificial intelligence and
pcrccptinn. A large part o l Nagy'a paper introduccd a
number of potential applications of pattern recognition
and the intcrplay bctwcen feature dcfini tion and the
application domain knowledge. Ile also emphasized thc
lincnr classificnticin mcthods; nonlincar techniques were
based on polynomial discriminant functions as well as on
potential fuiiclims (similar to what are now callcd the
kernel functions). Ey the time Kanal wrote his survey
paper, inore than 500 papers and about hall' a doxen
books on pattwn recognition were already publishcd.
Kana1 placed less emphasis 011 applications, but tnorc on
modeling and design of pattern recognition systems. 'Chc
discussion on automatic feature cxtractioii in 1891 was
bascd on various diskancc mcasurcs betwccn classconditinnal prnbability dcnsity functions and thc resulting error bounds. Kanal's review also contained a large
section on structiiral methods and pattern gratnmars.
In cnmparison to the state of the pattern recognition field
as dcscribcd by Nagy and Kanal in thc 1960s mid 1970s,
today a number of commercial pattern recognition systems
are available which even individuals can buy for personal
usc (e.g., machinc printed cliaractcr rccognition and
isolated spokcn word rccngnition). This has bcen made
possible by various technological developments resulting in
the availability of inexpensive sensors and powerftil clesktop computers. Tlw field of pattern recognition has become
so large that in this review we h a d to skip detailed
descriptions of various applications, as well as almost all
thc prnccdurcs which inorid domnin-syccific knowledge
(e.g., structural pattern recognition, and riile-based systems). l'hc starting point of our review (Secticin 2) is the
basic elements of statistical methods for pattern recognition.
It slirnkl be apparent that a halure vechr is a ruprcsentation of real world objects; the choice of the reprcscntatim
strongly influences the Classification results.
1. Its scctrnd crlition by Durln, Ilart, and Stork [43] is in prcss.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22,
8
NO. 1 , JANUARY 2000
TABLE 3
Links Between Statistical and Neural Network Melhods
The topic of probabilistic distance measiirw is currently not as important a s 20 ycars ago, sincc it is very
d i Cficult to cs tima te d ensity functioils j n high dimensional
feature spaces. Instead, thc crmplexity of classihation
procedures and the resulting accuracy have gaincd n
large intcrcst. The curse of dimcnsioiiality (Seclinn 3 ) R S
well ss the danger of o w r h i i i i n g are some of t h u
consequences of a complex classifier. It is now iindcrstood that thcvc problems can, to some extent, bc
circumvented using regularization, o r can even bp
complctcly resolved by a proper: design of classification
procedures. 'The study of suppoit vcctor machines
(SVMs), discussed in Section 5, has largely contributcd
to this uiiderstaizdii?g. In many real world prnblcms,
piitteim arc scattcrorl in high-dimensional (often) tionlinear subspaces. A s a consrqueiice, nonlinear procedures
and subspace appmachcs have become popular, both for
dimensionaliiy rcdiction (Section 4) m d for building
classifiers (Scckion 5 ) . Neural networks offer powerful
tools for tlicsc purposes. It is now widely accepted khat
110 singlo prticcdiire wil I complctcly solve a complcx
classification problum. There arc many admissible approaches, each capable of discl-iminahg patterns in
certain portintis o f the feature spacc. Tlic combination of
classifiers hcis, tliercforc, bucome ii heavily studicd topic
(Section 6). Varioiia approaches to estimahig thc error
rntc of a classifier are prcscntcd iiz Section 7. Thc topic of
unsupurvised classification 01' c1,ustering is covcrcd in
Section 6. Finally, Section 9 identifics the frontiers nf
pattern recogn i Lion.
It is o u r goal that most parts o f the paper can bc
appreciatcd by a iiewcomci: to the field of pattcm
test
Preprocessing
pattern
4
training
I
Preprocessing
pattern
Fig. 1 , Model for statislical pattern recognition.
t
-i
recognition. To this purpose, we h a w included il riiirnbw
of exainplcs to illustrate the pcrformance of v a rioiis
algorithms. Ncvcrtlwless, w e realize that, due to space
limitations, we h a w not been able to introduce all the
concepts complelcly. At these places, we havc to rely on
the background knowledge which tnay bc available only
to the more expericnccd rcndcrs.
2
STATISTICAL
PATTERNRECOGNITION
Statistical pattcrn recognition has bccn used successfully to
design n nuinbcr of commt?rcial rccogiiition systems. Tn
sfntisticd pattern rcctignition, ii pattern is rcprcsented by a
set of d features, or attributes, viewed as a hdimcnsional
feature vector. Well-known concepts from stalisticnl
decision klieory are utili zed to cstablish dccisioii boiindarics
bctwccn patl-orn classes. The recr)gnikion system i s operated
in two modes: training (learning) and classificatjor~(t~sting)
(see 15g. 1). The role of the prcprocc'sging module is to
scgincnt the pattern of intcrcst froin the background,
remove n t k c , normalize the pa ttcm, and any other.
operation which will contributc in defining a compact
representation of the pattern. I n thcl training mode, the
feature extraction/sclcction module finds thc appropriate
features for represenling the input patterns and the
classifier is trained
partition the feature space. The
Cccdback path allows a dcsigncr to optimize the preproccssing and feature extr~c~ion/xelectioi7strategies. 111 the
classificatioii mode, ihc h i n d classifier assigns thc input
pattern to one of thc pattcrii classes under considcration
based on the mcasurcd features.
Fc alU1'c
McasLircrllcll~
I~cat11sc
Extraction/
Sclcclion
t
ClassiTicalion
I
b
]-calming
JAlN ET AL.: STATISTICAL PATTERN RECOGNITION:A REVIEW
9
Thc decision making process in statistical pattern supervised nonparametric method which constructs a
recognition can be summarized as follows: A givcn pattern decision boundary.
Another dichotomy in statistical pattern recognition is
is tu bc assigned to one of c categories w1, wa. ,wE based
on a vcctor of d feature valuics x = ( x 1 , x 2 ~ ~ ~ ~Thu
, x r ~that
) . of supurviscd learning (labeled training samplcs)
features dre assumed to h a w a probability density or mass vcrsus unsupervised learning (unlabeled training sam(dcpcnding on whether the features are continuous or ples). The label OC n training pattern represents the
discrctc) function conditioned on the pattern class. Thus, a category to which that patkrn bclongs. In an unsuperprlttcrn vector z belonging t o class
is viewed as a n vised Icariiing problcin, sometimes the ntiinber of classes
observation drawn randomly from the class-conditional must be lcarncd along with the structure of each class.
probability function p(zlw,). A number of well-known The various dichotomics that appear in statistical pattcrii
decision rules, including the Bayes decision rule, the recrypition are shown in the tree structure of Fig. 2. As
maximum likclihood rule (which can bc vicwed as a we traverse the tree from top to bottotn arid left to right,
particular c a w of the Bayes rule), and thc Neyman-Pearson less iriformntion is available to the systcm dcsigncr and as
rule are available to define the decision boundary. The a result, the difficulty of classification problems jtzcreascs.
"optimal" Bayes decision rule for minimizing thc risk In some scnsc, most of the approaches in statistical
(expected value of the loss function) can be stated as pattern recognition (leaf nodes i n the tree of Fig. 2) are
follows: Assign input pattern 5 to class wi for which the attempting to implement tlic Bayes decision rule. The
field of clustcr analysis cssmtially deals with decision
conditional risk
making problems in the nonparametric and unsuperviscd
learning mode [81]. Further, in duster analysis the
nmnber of categories or clusters may not even be
specified; the task is to discover a reasonable categorixais minimum, where L ( w f , u j )is the loss incurred in deciding tion of the data (if m e exists). Clustcr analysis algorithms
ut when the true class is uj and P(wjlz) is the postcrinr
along with various tcchniquus frir visualizing and projectprobability [MI. In the case of the O/Z loss functinn, as ing multidimcnsional data are also referred to as
dcfincd in (2), the conditional risk becomes the conditional expiumtory d a h aiinlysis methods.
prubability of misclassification.
Yet another dichotomy in statistical p a k r n recognition
can
bc bawd 011 whether the decision boutidarics arc
L(q,LJj) = 01,, ii #=j ' j
obtained directly (geometric approach) or indircctly
(probabilistic density-based approach) as shown in Fig. 2.
For this choice of loss function, the Bayes decision rule can The probabilistic approach requires to cstimatc density
be simplified as follows (also callcd the maximum a functions first, and then construct the discriininant
posteriori (MAP) rule): Assign input pattern 5 to class wi if functions which specify the decision boundaries. On the
other hand, the geometric approach often constructs t11c
I'(walx) > P(wjlz) for a l l j # i .
(4 decision boundarios directly from optimizing certain cost
Various strategies arc utilized to design a classifier in functions. Wc should point out that under certain
statistical pattern recognition, depending on the kind of assumptions on the density functions, the two approaches
information available about the class-conditional densities. are equivalent. We will sec' cxamplcs of each category in
I f all of the class-conditional densities are conipletely Seckiun 5.
No matter which classification or decision rule is used, it
spccificd, thcn the optimal Hayes decision rule can be
used to design a classifier. Howcver, the class-conditional must be trained using thlc availablc training samples. As n
densities are usually not known in practice and must be result, the pcrformance of a classifier depends on bokh the
learned from the available training patterns. If the form of numbcr of available training samples as well a s the spccific
thc class-conditional densities is known (e.g., multivariate values of the samples. At the same time, thc goal of
Gaussian), but some of the parameters of the densities designing a recognition system is L o classify fukire test
(e.g., mean vcctrirs and covariance matrices) arc un- samples which arc likcly to be different from the training
known, tlwn wc have a parametric decision problem. A samples. Therefore, optimizing a classifier to maxiinine its
common strategy for this kind of problem is to rcplacc performance on the training set may not always result in thc
the unknown paramctcrs in the density functions by their desired perforinancc on a tcst set. The generalization ability
estimatcd values, resulting in the so-cnllcd Bayes plug-in of a classifier refers to its performance in classifying test
classifier. The optimal Bayesian strategy in this situation patterns which were not used during thc Iminiilg stage. A
requires additional information in the form of a prior poor generalization ability o f a classifier can be attributed to
distribution on thc unknown parameters. If Ihc form of any one of the following factors; 1)the number of features is
thc class-conditional densities is not known, then we tuu lnrgc relative to the number of training sainplcs (curse
operate in a nonparametric mndc. In this case, we must of dimensionality [80]), 2) the numnbcr of unknown
either cstimntc the density function (e.g., I'arzcn window parameters associatcd with the classifier is lnrgc
approach) or directly construct the decision boundary (e.g., polynomial classifiers or a large neural nctwork),
based on the training data (e.g., k-nearest neighbor rule). and 3) a classifier is too intensively optimizcd on the
In fact, the multilayer perceptron can also be vicwcd as a training set (nvertrainad); this is analogous to the
I
{
E E E TRANSACTIONS ON PATrERN ANALYSIS AND MACHINE INTELLIGENCE.
10
VOL. 22. NO, 1, JANUARY 2000
Class-Conditional
Dsnsiltes
I
Known
/.---
\\ 1
Unknown
, -
:f
\
Bayes Decision
Theory
Supervised
Unsupervised
Learning
\
Parametric
i\
Learning
Parametric
Nonparametric
Nonparametric
!
.'11, Decision
-
c,t,r
I /
Density-Based Approaches
Fig. 2.
iI
I
I?
Resolving
Analysis
LAJ''~
Geometric Approach
Various approaches in stalistical pattern recognition.
phenomenon of overfitting i n regression whcn there are too
many free parameters.
Overtraining has been investigated theoretically for
classifiers that minimize the apparent crror ratc (the error
on the training set). The classical studies by Covcr [33] and
Vapnik [162]on classifier capacity and coinplexity provide
a gond understanding of the mechanisms behind
overtraining. Complex clnssificrs (e.g., those having many
independent paramctcrs) may have a large capacity, i.e.,
they are able to represent many dichotomies for a given
dataset. A h q u c n t l y used measure for the capacity is thc
Vapnik-Chervonenkis (VC) dimensicmality 11621. These
results can also bc used to prove some interesting propcrties, for example, the consisteiicy of certain classifiers (see,
Devroye et al. [40],[41]). The practical use of the results 011
classifier complexity was initially limited because the
proposed bounds on the required number of (training)
samples wcre too conservative. Jn the recent dovclopmcnt
of support vector machines [162], however, these results
have proved to be quite useful. Thc pitfalls of overadaptation of estimators to thc given tmining set are
observed in several stages of a pattern recognition systcm,
such as dimensionality reduction, density estimation, and
classifier design, A sound solutioii is to always use a n
independent datasct (tcst set) for evaluation. In order to
aroid the necessity of having several indcpcndcnt test sets,
estimators arc oftcn based on rotated subsets of the data,
preserving different parts of the data for optimization and
evaluation [3.66]. Examples arc the optimization of the
covariancc cstimatcs for the Parzen kernel [76] and
discriminant analysis [hl], and thc iisc of bootstrapping
hir designing classifiers 1481, and for error estimation 1821.
Throughout the paper, stimc o f the classification methods will bc illustrated by simple experiments on the
following tlirec data sets:
Dataset 1: An artificial dataset consisting of two classes
with bivariatc. Gaussian density with the following parameters:
The intrinsic overlap between these two densities is
12.5 pcrccnt.
Dataset 2: Iris dataset consists of 150 four-dimcnsional
patterns in three classes (50 patterns each): Iris Setosa, Iris
Versicolor, and Iris Virginica.
Dataset 3: The digit dataset consists of handwritten
numerals ("W'-''9'') cxtracted from a collection of Dutch
utility maps. Two fiundrcd patterns per class (for a total of
2,000 pattenis) are available in the form of 30 x 48 binary
imagcs. 'I'husc characters are represented in terms of the
following six feature sets:
I.
2.
3.
4.
5.
6.
76 Fourier coefficients of the character shapcs;
216 profile correlations;
64 Karhunen-Lohe coefficients;
240 pixel averages in 2 x 3 windows;
47 Zernike moments;
6 morphological features.
JAlN ET AL.: STATISTICAL PATTERN RECOGNITION: A REVIEW
11
Details of this dataset are availablc in [IhOj. In our imum discrimination between the two classes. 'l'he only
experiments we always used the same subsct cif 1,000 parameter i n the densities is the incan vcctor,
patterns Cor tcsting and various subs& of the remaining m = ml . -m2.
Trunk considered thc hllowing twr) CRSC'S:
1,000 pntterns for training.' Throughotlt this paper, when
we refer to "the digit dataset," just h e Kxh:htinen-Loeve
I. Thu mean vactor m is known. In this situation, we
Icaturw (in item 3 ) are meant, unlcss stated otherwise.
can use the optimal B a p dccisirm rule (with a U/1
loss function) tu construct the decision boundary.
AND PEAKING
3 THECURSE OF DIMENSIONALITY
The probability of error as a function of d c m be
cxprosscd as:
PHENOMENA
7
The performance of ;1 classifier depends on tlic inlcrrclationship bctwccii sample sizes, nuinbur of features, and
classifier complexity. A naive table-I ookup tcchniqiie
(partitioning the feature space into cells and associating a
[t is easy to vcrify that liiii(l,Do Pt,(d)= 0. In other
class label with each cell) requites the numbcr of training
words, we can perfectly discriminate the two clnssus
data points to be an exponcntial function of thc fcaturc
by arbitrarily increasing h e nuiiiber o f features, d.
dimension [IS]. This phenomenon is termed as "curse of
2. The mean vector m is unkntiwii and n labeled
dimensionality," which lcads to the "peaking phcnnmcnon"
training samples arc available. Trunk f o m d the
(scc discussion below} i n classificr design. It is well-known
maximum likelihood cstiiiiatc f i of m and used thc.
that the probability of misclassification of a decision rule
plug-in decision rulc (substihte fi for n i in the
dncs no( incrcase as the number of fcatures increases, as
optimal Rayes decision rulc). Now Ihc probability of
long a s the class-conditional densities are completely
error which is a function of both n and d can be
known (or, equivalently, thc number of training samplcs
written as:
is arbitrarily large and representative oC thc undcrlying
dcnsitics). However, it has been often observed in practice
that the added features inay actually degrade the pcrfnrmance of a classifier if the izumber of training samples that
are used to design h o classifier is small rclativc to the
number of features. This paradoxical behavior is rcfcrrcd to
as thc? peaking phenomenon3 [SO], [131], [132]. A simple
explanatinn f r r h i s pliei~oinenoni s as follows: The most
commonly used parametric classifiers estimate thc unTrunk showed that liind. ,m cl(n,rE)
=
which implies
known parameters and plug thuin in for the true parameters that the probability of error approaches tlw maximuin
in tlic class-coi~ditioni~l
densities. For a fixcd sample size, as possiblc vnluc of 0.5 for this two-class problem. This
the number of features is incrcased (with a corrcspnnding demonstrates that, unlike case 1) we cannot arbitrarily
iiicrcaso in the number of unknown parameters), the incrcasu tlihc number of features when the paramctcrs of
reliability of the prranieter estimatcs decreases. Cotwe- class-conditional densities arc estimated from il finitc
qiiently, the performance of the resulting plug-in classifiers, number of training samples. 'I'he practical implication of
for A fixed sample sizc, may degrade with a n iiicreasc in thc the ciirsc of diiiiciisionality is that a system designer should
number of features.
try to selecl only a small number of salient features rvheii
Trunk [I571 provided a siinple example lo illustrate the confronted wi.th a limited training set.
curse of dimensionality which we reproducc bcluw.
All of the commonly uscd classifiers, including multiCoiisidcr the two-class classification problem with equal layer feed-forward networks, can suffer from the curse oE
prior prcibabilitics, and a d-dimensional multivariate GFILIS- dimensionality. While an cxact relationship between thc
sian distribiition with the idciitity covariance niatrix foi. probabiliky of misclassification, the number of training
cach class. The mean vectors for tlw two classes havc the samples, the numbcr of featurcs and the true paramctcrs of
foilowing compmcrits
thc class-conditional densities is very difficult to establish,
some guidelines havc been suggested regirding the ratio of
1 1
1
7111 =(I.,allcl
tlic sample size to dimensionality. It is gwcrally accepted
Jd
that using at lenst teen times as many training samples per
class as the number of fccnturcs ( n / d > 10) is H good practice
to follow i n classifier dcsipi [SO]. The more complex the
Note that the features arc statistically jndependcnt and the classifiiur, thc larger should the ratio o f sample size t o
discriminating power of Lhc successive features decreascs dimensionality bc to avoid the curse of ditnensimality.
monotonically with the fjrst featurc providing the max-
4,
4
DIMENSIONALITY
REDUCTION
2.'l'hc dnhsct is nvoilahlc thmugh the University o f Cnlihriiia, Irvinc
Machine Imrning Kcpositor)- ( ~ v ~ v w . i ~ ~ . i ~ c i . ~ ~ i i / - m l ~ ~ ~ r n / M L
l ~ c ~ oarc
s i t otwo
~ - main reasons to keep the diiiicnsionality c1f
There
y.html)
the
pattern
representation (i.e., the number of featiircs) as
3. [CIthc rest of Ihin pipw, UT do nut makc distinclion be!rvc.cn tlw cul-sc'
small as possible: measurement cost and classificalicni
of diniensiw?nlity and thc pt?Jking phenomenun.
12
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO, 1 , JANUARY 2000
accuracy. A limited yet salienl-fuaturc set simplifies both the
pattern representatioii and the classifiers that are built on
the selected representation. Consequently, the rcsulting
classifier will be faster and will iisc less memory. Moreover,
as stntcd earlier, a small number of fcaiurcs can alleviate the
curse nE dimcnsionality when the number of training
samples is limited. On lhc othcr hand, a reduction in thc
nuiiibcr of fcatures may lead to a lnss in thc discriinination
power and tliercby lower the accuracy of the resulting
rr?copiition system. Watanabe's rrgly d u c k h g tliromm [I631
also supports the need for a careful choice OF thc features,
since it is possible to makc two arbitrary patterns sirpdnr by
encoding them with a sufficiently large number of
redundant features.
It is important to make a distinction betwccii fcaturc
selection and feature extraction. Thc term fcahirc selection
refers to algorithms that sclect the (hopefully)best subsct of
the input feature set. Mcllmds that create new feattires
based on trausformatioiis or combinations of the original
fcature set we called featurc extraction algorithms. However, the terins feature selection and fcahirc extraction are
uscd intcrchangcably in the literature. Note that ofton
feature extraction precedes feattirc selection; first, features
are extracted from the s m w d data (e.g., using princi.pal
component 01' discriminant analysis) and then some of the
extractcd fcaturos with low discrimination ability arc
discarded. The choice between fcaturc selection and feature
cxtraction depends on thc application domain and the
specific training data which is available. Feature scleciioii
lcads to savings in measureinclit cost (since some of the
features are discarded) arid the selected featurm rotain their
original physical interpretation. In add ition, the retained
features may be important for mderstandjng the physical
process that generates thc pnttcrns. On the other hand,
transformed features generated by fcaturc cxtraction inay
provide a butter discriminative ability than thc best subsct
of given features, but these new features (a linear o r a
nonlinear combination of given features) may not h a w a
clcar physical meaning.
111 many situations, it is useful to obtain a hvo- or tlzreedimcnsional projection of the given multivariate data (n x ri
pattern matrix) to permit n visual examination of tlze data.
Several graphical techniques also exist for visually observing multivariate data, in which the objective is to exactly
depick each pattern as a picture with d degrees of freedom,
where d is the given numbcr of features. For example,
Clzernoff [29] represents cadi pattcrn as a cartoon face
whose facial characteristics, such as nose lcngkh, mouth
curvaturrc, and eye size, are made to correspond to
individual features. Fig. 3 shows three faces corresponding
to the mean vectors of Iris Setosa, Iris Versicolor, and Iris
Virginica classes in tlie Iris data (150 four-dimcnsional
patkriis; 50 patterns per class). Note that the face associated
with iris Setosa looks quite diflercnt from the othcr two
faces which implies that the Setosa catugory can bc well
separated from the remainiiig Lwo catcgnrics in the fourdimensional feature space (This is also evident in the twodimensioiial plots of this data in Fig. 5 ) .
The main issue in dimensionality reduction is the chnicc
of i\ criterion function. A commonly used criterion is the
Classification error of a fmturc subsct. But thc classification
error itself cannot be reliably estimated when the ratio of
saniple size to tlie iitimber of features is small. In addition to
the ch(iico of R critcrion function, wt. also nccd to determine
the appropriate dimensionality of tlic rcduccd fcnturo
spwe. The answer to this question is embedded in thc
notion of tlw intrinsic dimcnsionality of data. Intrinsic
dimensionality essentially determines whether tlic givcn
rl-dimcnsioiial pattcrns can be described adequately in rl
subspace of dimensionality less than (I. Pnr cxample,
d-dimcnsional patterns along a reasonably smooth curve
have an intrinsic dinicnsioiiality of uiic, irrcapective of the
value of d. Note that the intrinsic dimensionality is not tlic
samc as thr! lincar dimcnsionality which is a globa I property
of the data involving tlic iiunibcr of significant eigenvalues
uf thc covariance matrix of tlze data. While several
algorithms are available to estimate the intrinsic dimensionolity [MI, they do not indicate how a subspnce of the
identified ditnensionality can be easily identified.
We now briefly discuss some of the commonly uscd
methods for feature extraction and feature selcction.
4.1
Feature Extraction
Vcaturc cxtrnciion methods determine an appropriate subspace of dimensionality
(cithcr in a lincar (ir il nonlinear
way) in the original feature space of djmensionality d
{rri 5 d). Linear transforms, such as principal component
analysis, factor analysis, li ticap discriminant analysis, and
projuction pursuit have been widely used in pattern
recognition for fcaturc extraction and diinensioiiality
reduction. The best known linear feature extractor is thc
principd component analysis (PCA) or Karhunen-Lo&ve
expa tision, that cr)mputccs thc na largest eigenvectors of the
d x d covariatice matrix of the 71, d-diincnsirinal pattt'riis. The
linear transformation is defined as
Y
XU!
(7)
whew X is thc givcn n. x d pattcrn matrix, Y is the derived
II x rii pattern matrix, and 1-I is thc d x ~nmatrix of Iincnr
tmnsftirmation whose columns are the eigenvectors. Since
I T A usc's the mnst cxprcssivc fciltures (cigeiivectors with
the largcs t eigenvalues), it effectively approximates the data
by a lincar subspace using the incan squared m o r criterion.
Other methods, like projection pursuit [ 5 3 ] and
independent component analysis ([CA) [31], [:[I], [XI,1961
art! mort appropriate for non-Gaussian distributions since
they do not rdy on thc second-ordcr propcrty of the data.
ICA has been successfully uscd for blind-sourcc scpmation
[781; ex tracti rig U nea 1: feat ii re comb i nations that dcfiiw
indcpcndcnt sources. This demixiiig is possible if at most
one of the sources has a Gaussian dishibulion.
Whereas PCA is a n unsuporvisod lincar feature extraction method., discriminant analysis uses thc category
information associated with each pattern for. (linearly)
extracting the must discriminatory fcaturos. In discriminant
analysis, interclass separation is emphasized by replacing
the total covariance malrix in PCA by R gciicrd separability
measure like the Fisher criterion, which rcsulls in finding
thc cigcnvcctors of j91,;'Sb(the product of the iwerse of the
withimclass scatter matrix, S,,,, aiid the behueen-class
JAlN ET AL.: STATISTICAL PATTERN RECOGNITION: A REVIEW
13
i
Setosa
Versicolor
.. . .
Virginica
Fig. 3, Chernoff Faces corresponding to the mean vectors of Iris Sstosa, Iris Versicolor, and Iris Virginica.
critcrioii is the stress funchin introduced by Saiiimon
['143.] and Nicmanii [1:14]. A problem with MDS is that it
does not give an explicit mapping function, s o it is not
passible to place a ncw pattern in a map which has been
computed for a given training sot without repeating the
mapping. Several techniques havc bwn investigated to
address this dcficicncy which range from liiicar interpolation to training a neural network 1381. It is also possible to
redefine the MDS algorithm so that it directly produccs a
map that iiiay be used for new test patterns [165].
A fwd-forward neural network offers an integrated
procedure for fcaturo extraction arid classification; the
rmtput of each hidden layor may be interpreted as a set of
new, often nonlinear, features presented to thc output laycr
K ( . E , ~ )iT?(z) (I)(?))
for classification. In this sense, multilayer networks serve as
As a result, thc kernel space has c? well-Jefincd metric. feature extractnrs [ml.For cxample, the networks uscd by
Examples of Merccr kernels include ptli-order polyncmiinl Fukushima [62] c t al. and Le Ciin et al. 1951 h a w the so
called shared weigh[ layers that arc in fact filters for
(x - y)" and Caussian kernel
extracting features in two-dimensional images. During
lii2.,
training, tlie filters are tuned to the data, so as to innxiinize
(.'
r:
the classification performance.
Lct X be the normalizcd n, x (Ipattern matrix with zero
Neural networks can alsti bv used directly for icature
mean, and +(X) be the pattern matrix in the J' space. extraction in an unsupervised mode. Fig. 4a shows thc
Thc linear PCA in the I.' spncc solves the eigcnvcctors of the architecture of a nctwork which is able to find the PCA
which is also callcd tlw subspacc 11171. Instead of sigmoids, the neurons have linear
correlation matrix @(X)iP(.X)~",
kcrncl matrix K ( X , X ) . In kernel PCR, the first m transfer fiinctions. This network has d inputs and il outputs,
eipnvoctors of K ( X , X ) arc cibtained to define a transfoi-- where d is Lhc givcn number of featmcs. The inputs are also
mation matrix, E. ( E has size ri x m, where T I L represents thc used as targets, forcing the output layer to reconstruct ttic
desired number o f features, ' ~ 51 d). Ncw patterns 2 arc input space iisiiig cmly one hidden layer. l'hc ihrce nodes in
niappcd by I < ( x !X)IJ, which are now represented relativc the hidden layw capture tlie first three principal compoto the training sct and not by their measured feature valucs. nents [MI. If two nonlinear layers with sigmoidal hidden
Note that for a cornplclc rupresentation, up to TI eigenvcc- w i t s arc also iiicluded (see Pig. 4b), then a nonlitwar
tors in I? m a y be needed (depending on the kerncl function) subspace is found in the middle layer (also called lhc
by kcrncl PCA, while in lincar PCA a sot of d eigenvectors bottlcncck layer). The nonlinearily is limited by the size of
represents thc original Featurc space. l.Iow thc kernel these addilioniil layers. These sn-callcd autoassociative, or
function should bc chosen for a givcn application is still nonlinear I T A nchvorks offer a powerCul tciol to train and
ail open issue.
describe nonlinear subspaces [98]. o j a [118] shows how
Mtiltidimeiisional scaling (MDS) is anolhcr nonlinear autoassociative networks cnii be used for TCA.
fcattire extraction tcchniquc. It aims tu rcprcseiit a multiThe SelLOrgnni~ingMap (SOM), or Ihhonen Map 1921,
diniensional dataset in trvo or thtcc dimensions such that can also be used for nonlincar feature extraction. In SOM,
the distance matrix in the original d-dimensional featiirc iieiirons arc arranged in an m-dimensional grid, where 1'11. is
space is prescrvcd as faithfully as possible in the projected usually 1,2, OF 3. Bach iieuron is comwctcd to all the d input
space. Various strcss functions are uscd hir measuring the units. The weights on the connections for each m m o n form
ycrformance of this mapping [20]; ttic most popular a d-dimensional weight vcctor. Ditring training, patterns arc
scatter matrix, S b ) [58]. Another superviscd criterion for
non-C:aussiaii class-conditional densities is bawd on the
Patrick-Fisher distancc using Parxen density estimates 1411%
There are swcral ways to define nonlinear Featurc
extraction tcchniques. One such method which is directly
related to PCA is called the Kerncl PCA [73], [145]. Tlic
basic idea of kerncl PCA is to first map input data into s(mc
new featurc spacc F typically via a nonlinear runctim <I)
(c.g., polynnminl o f degree p) and then perform a lincar
PCA in the mapped spacc. Huwever, the F' space often has
a very high dimension. To avoid computing tlie mapping @
explicitly, kernel PCA empliiys only M:ercer kernels which
can be decomposcd into a dot pi:oduct,
14
IEEE TnANSACTlONS ON PATTERN ANALYSIS AND MACHlNE INTELLIGENCE, VOL. 22,
NO. 1,
JANUARY 2000
Fig. 4. Autoassociative networks for finding a three-dimensional subspace. (a) Linear and (b) nonlinear (not all the connections are shown).
prcscnlcd to Ihc iictwork in a random ordcr. At each
presentation, the winner whose weight vector i s the closest
to Ihc input vcctor is first idcntificd. Thcn, all thc neurons in
tlw ncighborhond (dclincd on Ihc grid) o f thc winiior are
updated such that their rvejght vectors m o w towards thc
input vcctor. Conscqucntly, after training is done, the
weight vectors of neighboring neurons in the grid arc likdy
to repwscnt input patterns which are close in tlze original
feature space. Thus, a "topology-prcservir~~"m a p is
h m c d . When tlw grid is plotkd in the original spncc, thc
grid coiinectioiis are more or less stressed according to the
density of the training data. Thus, SOM ciffcrs a n
,/ri-diineiisioiial map with a spatia1 connectivity, which can
bc iritcrprctcd as fcalurc cxtractim. SOM is different from
learning vector quantization (LVQ) because no neighborhood is dclincd in LVQ.
Table 4 summarizes the feature extraction and projection
inrthods d iscussed ilbove. Note that the adjective nonlinear
may bc uscd bcith for tlw mapping @cing n nonlinear
function of the original features) as well as for the criterion
{urictirm (for non-Gaussian data). Fig. 5 shows an Pxamplc
of four different two-diinensioiznl projections of the fourdin~cnsionalIris datasct. Fig. 5a and Fig. 5b shc~~.v
two lincnr
mappings, while Fig. 5c and Fig. 5d depict two nonlinear
mappings. Chly thc Pishcr mapping (Fig. 5b) makcs iisc of
thc category information, this being the main reason why
this mapping cixhibits the best scparation bctwccn thr! thrcc
categories.
4.2 Feature Selection
Thc prriblcrii of feature selection is defined as follows: giveti
a set of d features, select n subset of size ,/rL that Icads to the
smallest classilicaiirm error. 'Thcrc has bccn a rcsurgcncc of
interest in applying feature selection methods due to the
largc nuiiibcr of fcalurus uncrnmtcrcd in tlw k)llowing
situations: 1 ) multisensor fusion: features, computed froin
different sensor modalities, are concntcnatcd to form R
feature vector with a large II timber of co~npoiients;
2) integration of multiple data models: seiisor data can be
modeled using different approaches, where the model
parameters serve as features, aiid tlw ynranwte1.s from
different inodcls can bc poolcd to yield x high-ciiincnsioiial
fcnturc vector.
Lct Y bc thc givcii sct of fccahircs, with cardinality rl and
k t nt reprusent the desird number of features in the
selected subset X , X r: Y . Let the featlire solectioiz criterion
funclion for h c sct X be rcprcscntcd by J ( X ) . Ect LIS
assitme that a higher value of #J indicates a better feature
subsot; a natural choicc f o r tho critcricm function is
.I - (1 l < , )whcrc
,
i<. dcnotes ihc classification crror. Ttic
use of Pc in the criterion function .makes feature selection
procedures depcndent on the specific clnssificr that is usud
and thc sizcs of thc training and test scts. Thc most
straightforward app ronch to the featii re selection problem
would reqiiirc 1) cxnmining all
possible stdsets of s i ~ c
mn,, and 2) sclcciing thc subsot with thc largc'st valui: of .7(.).
However, the nuiiibcr of possiblc subscts grows combimtorially, making this cxliaustivc scarcli impractical for cvcn
moderate values of 'm and rl. Cover. a n d Van Camyenhout
[ 3 5 ] showed that no nonexhaustive scqiiential feature
selectim procedure c a n be giiaranteeci to prnducc thc
riptiinal siibsct. T h y fur1hi.r showcd that any ordering o f
the classification errors of each of the 2" feature subsets is
possiblc. Tlicrcforc, in ordcr to giiarnntcc h c optimality of,
say, n 'IZdiir~ensjonalkature subset out of 24 available
features, approximately 2.7 inillion pussiblu subscts must bc
cvaluatcd. 'l'lic only "optimal" (in terms of a class ol
monotonic ci:itei:i on functio.ns) feature selection method
which iivoids the exhaustive search i s based on the branch
and bound nlgorihn. This prticcdum avoids an cxliaustivc
scarch by using intermudiatc results fur obtaining bounds
on the final criterion value. The key to this algorithiin is the
~
(3
15
JAlN ET AL.: STATISTICAL PATTERN RECOGNITION: A REVIEW
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,.
i
Fig. 5. Two-dimensional mappings of the Iris dataset (+: Iris Sstosa; *: Iris Versicolor; 0:Iris Virginica). (a) PCA, (b) Fisher Mapping, (c) Sammon
Mapping, and (d) Kernel PCA with second order polynomial kernel.
monotonicity property of the criterion function J( .); givcn
two features subsets X I and X z , if X1 c X2, then
J(XL)< J(Xy). In othcr words, the performancc of a
feature subset should improve wliencvor a feature is added
to it. Most commonly used criterion functions do not satisfy
this monotonicity property.
It has becn argued that since feature selectinn is typically
done in an off-line manncr, thc execution time of ii
particular algorithm is not RS critical as the optimality of
thc Ccalurc subset it generates. While this is true for feature
sets of intrderatc size, severill recent apylicatiuns, particularly those in data mining and document classification,
involvc thousands of features. In such cases, thc computational requirement o f a fcaturc selection algorithm is
cxtrcmcly important. As the number of fcature subset
evaluations may easily bccomo prohibitive for large fcaturc
sizes, a number of suboptimal selecticnl techniques have
been proposcd which essentially tradeoff the optimality of
the selected subset for computational efficiency.
Tnblc 5 lists most of the well-known fenturc selection
methods which have bem proposcd in the literature [85].
Only thc first two methods in this tablc guarantee an
optimal subsct. All other strategies are suboptimal due to
the fact that the best pair 01: Fcaturcs nccd not contain the
best single feature [34].Iiz general: good, larger katurc sets
do not necessarily iiicludc thc good, small sets. As a result,
thc simple method of selecting just the best individual
featitres may fail dramatically. It might still be USCFUI,
however, as a first step to selcck some individually good
features in decreasing very largc fcnturc sck (e.g.,hundreds
of fwturcs). Furtlier selection has to be dotic by more
advanced methods that take fcalurc depciidencies into
account. These npernk citlwr by evaluating growing fenlurc
sets (forward selectinn) by cvaluating shrinking fealurc
scts (backward selection). A siinplc. scqwntial method likc
SFS (SBS) adds (dclctes) one feature a t a titnc. More
sophisticated techniques arc thi. "Plus 1 - take away r"
strategy and the Sequential Floating Search methods, SFFS
and SRFS [:L26]. 'l'hcsc methods backtrack as long as they
find improvements compared to prcvious fmturc sets of thc
mint sim. In almost any large feature selection problem,
these methods perform better than the straight scqiicntial
searches, SFS and SBS. SFPS and SWS methods G i d
"ncstcd" sets of features that remaiii hiddcn otlierwisc,
but the nuinbcr of feature set evaluations, howcver, may
easily incrcast. by a factor of 2 to IO.
16
TABLE 4
Feature Extraction and Projection Methods
.
Mctlloil
Pra p er ty
Annlysis
. .
1’cii.jccLion Pursuit
(,WA,l)
[n addition to thc scarch strategy, the user needs to select
an appropriate evaluation criterion, J(.) arid specify the
value of ‘wt. Most featurc selection methods use the
classification error of a feature subset to evaluate its
cffectiveness. This could be donc, frir example, by a “-;I
classificr iising the leave-one-out method of error estimation. However, use of a different classifier and a diffcrcnt
mcthod for estimating the error rate could Iead Lo ii
different fuaturc subset being selected. Ferri et al. [50] and
Jain and Zongkcr [85] have coinpared several of the feature
TABLE 5
Feature Selection Methods
JAlN ET AL.: STATETICAL PATTERN RECOGNITION: A REVIEW
selection algorithms in terms of classification error and run
time. The general conclusion is that the sequential forward
floating search (SPlS) method performs almost as well as
h e branch-and-bound algorithm and demands lower
computational resources. Somol et al. [-I541 have proposed
a n adapthe versinn of the SFPS algorithm which has been
shown to have superior pcrformance.
The feature sclcction methods in l’ablc 5 can be used
with any of the well-known classifiers. hit, if a multilayer
fced forward network is used fm pattern classification, then
thc ntidc-priming method siinultancously determines both
tlic optimal feature subset and tlic optimal network
classifier [26], [103].First train a nctwork and then removc
the least salient nudc (in input or hidden layers). The
rediiced network is traincd again, followed by a removal of
yet another least salient node. This procedurc is repeated
until the desired trade-off between classification crror and
size of the nefwork is achicved. The pruning o f an input
node is eqii ivalent to rcmoving the corresponding fciiture.
How reliable are thc fcature selection results whcii the
ratio c ~ fthc available number of training samples to the
numbcr of features is small? Suppose tlie Mahalanobis
diskincc [SS] is used as tlie feature sclcction criterion. It
dcpcnds 011 the inverse of the average class covariance
matrix. The imprecision in its estimate in small sample size
situations can result in a n optimal feature subsc.1 which is
qiiite different from the optimal subset that would be
obtained when thc covariance matrix is knnrun. Jain and
Zongker [tis] illustratc this phenomenon for a two-class
classifka lion prriblcm iiivolving 20-diinensional Gaussian
class-conditional densities (the same data was also used by
Trunk [I571 to demonstrstc the ciirse o f dimmsionality
phcnomcnon). As expected, the quality of the selected
feature subsct for small training sets is poor, but improves
as Ihc Lrainiiig set size increases. For example, with 20
patterns in h c training set, the branch-and-bound algorithm sclcctcd a subset of 10 features which included only
five fcnturcs in common with the ideal subset of 1.0 features
(when densities were known). With 2,500 patterns in tlic
training set, the branch-and-hmd proced Lire sclcctcd a 10feature subset with only one wrong feature.
l3g. 6 shows an example ol: thc feature selection
prticcduro using the floating search tcchnique on the I’CA
fcaturcn in tlw digit dataset for two different training set
sizes. Thc kcst sct size is fixed at 1,Oflo patterns. In each of
tlw seleckcd €catitre spaces with dimcnsimalities ranging
from :L to 64, the Bayes plug-in classifier is designed
assuming Gaussian densities with equal covariance
malriccs and evaluated on the tcst w t . The feature selection
criticrim is thc minimum pairwise Mahalanobis distance. In
tlw sni~llsample s b e case (total of 100 training patlerris),
the cwse of d imcrisicninlity plienomenon can be clcarly
observed. In this caw, tlw optimal number of Icaliircs is
about 2U which equals n / 5 (?E = LOO), where n is thc nitmbcr
of training patterns. The riile-of-thumb o f lintring less than
r r / l O features is on tlw s a k side in general.
5 CLASSIFIERS
oncc a fcahirc sclcction 0 1 classification procediire finds a
proper representation, a classifier can be desigtwd using a
17
.,: ........................
. . . . . . ,,;:
. .:. .........
.
........................
.......
100 iraining patterns
.......
1000,!raining patterns
I .............................................
OO
10
20
I-
30
40
No. of Features
....... )., ..............
50
60
Fiy. 6. Classification error vs, the number of features using the floating
search feature selection technique (see text).
number of possible approaches. In practice, the choice of a
classifier is B difficult problem and it is oflcn based on
wliich classifier(s) happen to be available, nr bust known, to
tho iiser.
We identify three diffcrcnt appimches to designing a
classifier. ‘ l h simplost and the most intuitive apprtmch to
classifiicr design is based on the concept of similarity:
patterns that RE siniilar should be assigned tci the same
class. So, once a good inetric has been establishcd tu define
similarity, pattcrns can be classified by template inatching
or tlw ininimum distance classifier using a few prototypes
per class. The choice of thc metric and the prototypcs is
crucial to the success of this approach. I n thc nearest mean
classifier, selecting prntolypcs is very simple and robust;
t.acli pattem class is rcprcscnted by R single prototypc
which is the m e a n vector of all tlic training patterns in thnl
class. Mow advariccd tccliniques for compiitj ng grrrtotypes
are vector quantization [115], [I711 and learning vector
qunntixation [92], and the data rcduciinn mcthods associated with the orwilearcst nciglibor decision rule (:i -NN),
such as editing and colidensing [ 3 9 ] . Ttic ii-tost
straigtithrward 1-NNride can be convenieiitly uscd as a
benchmark for all tlic other classifiers sjnce it aplwars to
always provide R reasonable classification performance in
most applicntions. Further, as the I-NN classilier does not
require any user-spccificd parameters (except perhaps the
distancc nictric used to find the nearest tirighbnr, but
Euclidcaii distance i s commonly used), its classification
results arc implementation independclit.
In inany classification problems, the classificr is
expected to have s(mc desired iinwitIt1t propertics. An
example is the shitt invariancc Of characters it) character
recognition; a cliaiip in a character’s location should riot
affcct its classification. ‘If the preproccssiiig or tlic
representaticin schcme does not normaliec thc input
pattern for this invariance, then tlic samc character may
be rcprusented at multiple positions in thhc fcature spncc.
T11t.s~. positions define a nnc-diiiicnsiniini subspace. As
more invariants are considered, the dimeiisioiiality of this
subspace correspondingly iiicrcases. Template matching
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,
18
or tlie nearest mean classifier can be viewed as finding
the ncarcst subspace 11161.
The second main conccpt used for designing pattern
classifiers i s bascd on thc ivobabilistic approach. Thc
optimal Bayes decision rulr! (with the O j l loss function)
assigns a pattern to tlie class with the maximum posterior
probability. This rulc can bc modified to take into account
costs associated with different types of misclassifications.
1;nr known class coiiditionai densities, thc Bayes decision
rule givcs thc optiinum classifier, in the scnw hat, for
given prior pmbabilitics, loss function and class-conditional densities, no other decision rulc will have a lower
risk {i.e.,cxpoctcd value of the loss function, for example,
probability of error). If thc prior class probabilities are
cqud and a 0/3. loss function is adoptcd, the Bayes
decision rule and the maximum likelihood decision rule
exactly coincide. In practice, the enipirical Bayos dccision
rule, or "plug-in" rule, is used: the estimates cif thc
deiisiti.es are w e d in placc of the true densities. These
density estimates are either parametric or nunparametric.
Commonly used parametric modcls arc multivariate
Gaussian distributions 1581 for continuous Ccaturcs,
binomial distributions for binary features, and
mtiltinormal distributions for intcgcr-valued (and categorical) features. A critical issue for Gaussian distributions
is the assumption made about the covariance matrices. If
tlic covariance matrices for different classes arc assumcd
to be identical, then the Bayw plug-in rulc, called Dayesnrirmal-lincar, pi'ovides a linear decision boundary. On
the other harid, if the covariancc mati<icesare assumed to
bi! diffcrcnt, the resulting Hayes plug-in rulc, which we
call Bayes-izormal-quad ratic, providci a quadratic
decision boundary. In addition to the commonly uscd
maximum 1ikclihor)d cstiinator of the covariance matrix,
various regularization tcchniqucs I541 are available to
obtain a robust estimate in small samplc size situatiuns
and the lcave-one-ou t estimator is available for
minimizing the bias [76].
A logistic classifier [4],which i s based on the iliaximum
li kelihnod approach, is well suited for mixed data types. For
a two-class problem, the classifier maxiniizcs:
whcrc cl~(z:O)is the posterior probability of class L+, given
:U, 8 denotes the set of unknown parameters, and x,(j)
deizotes the ith training samplc from class L+, j = I , 2. Given
any discriniinant function U ( z ;e), whcrc 0 is tho parameter
vector, the posterior probabilitics can be derived as
which arc callcd logistic functions. For linear discriminants,
D ( x ; (8) can be easily optjmized. Equations (8) and (9)
m a y also bc used for estimating the class conditional
poskrior probabilities by optimizing D ( x ;0 ) over the
training set. The relationship between the discriminant
function U(z: 0) and the posterior probabilities can be
e),
Vol.. 22, NO. 1 , JANUARY 2000
dcrivcd as follows: Wc know that the log-discriminant
function for the Uayes decision rule, given the posterior
probabilities q , ( x ; O ) and qa(.x:O),is log(ql (rjO ) / q J ( x ; U ) j .
Assume that I)($; 0) can bc optimized to approximate the
Baycs dccision boiindilry, i.e.,
D(x:e)
= log(yliz;8)/~~2(2;8)).
(10)
We also h a w
+ qy(3::O)
ill (s;B)
= 1.
(11)
Solving (10) and (13.) for (z; 0) sncl q j ( x ; # )results in (9).
Tlic t w o well-known nonparametric decision rules, the
k-neatest neighbor (k-NN) rule mid thc Parzcn classifier
(the class-cunditionnl dcnsities iwe replaced by their
cstiinntcs using the Parxeiz window approach), whilc
similar in nahirc, give different results in practicc. 'L'hcy
both have essentially one free parameter each, thc nuinbor
of neighbors I;, or the smookhing paraiiictor of the Parzen
kemd, both of which can be optimized by a leave-one-out
estimate of the error ratc. Further, both lhcsc classifiers
rcquirc tho coiiiputation of the distances between a test
pattern and all the patterns in thc training sck. Thc most
convcnicnt way to avoid those large numbers of cornputations is by a systematic reduction of the training set, e.g., by
vector: quantization tech nicptes possibly combincd with an
optimized metric or kcrncl [613], [hl]. Other possibilities like
table-look-up and br-anch-and-bound mclhorls I421 arc less
efficientfor largc dimcnsionalitics.
I ' h third catcgriry o f classificrs is to coiistruct decision
boundarics (gctiinctric approach in Fig. 2) directly by
optiniizing certain error criterion. While this approach
deponds on the chosen mctric, soiizetinzes classifiers of this
type may approximate the Bayes classifier asymptotically.
'I'hc driving fwcc of the training procedure is, however, the
minimization of a criterion such as the apparent classificntion error (ir thc iiican quarcd error (MSE) between the
classifier output and some preset target valuc. A classical
example of this typc of classificr is Fisher's linear
discriminant khat miniinizcs thc MSE bctween the classifier
output and the desired labels. Another examplc is thc
single-layer perceptron, where thc scparaLirig hypcrplaiic is
iteratively updatcd as a luiictioii of thhc distances of the
misclassified patterns from the hyperplane. If thc siginnid
function is used in combination with thc MSE criterion, as
in fccd-forward iieiiral.nets (also called niultilayer percepirons), the perceptron may show a bchavior which is siinilir
io other liiicar classificrs [133]. It is important to note that
netird networks themselves c a n lcad to inany diffcrcnt
classificrs dqwnding on how they arc trained. While the
hiddcn layers in multilayer perceptroils alIow nonliiicar
dccision boundaries, they also increase the daiigor of
ovurlraining tlw classificr sincc the number of network
paramctors incrcascs as more layers and more neurons pcr
layer are added. Thercfnrc, thc rcgularization of neural
networks may bc ncccssary. Many rcgulariLation mechanisms arc alrcady built in, such a s slow training in
combination with early stopping. Other rcgularimtion
mctlwds include the addition of noise and weigh1 decay
[Is],[28], [:137], and also Baycsian Icarning 1:113].
19
JAIN ET AL.: STATISTICAL PATTERN RECOGNITION: A REVIEW
One of thc interesting charactcristics of multilayer
pcrccptrons is that in addition tu classifying an input
pattern, they also provide a confidence in the classificntion,
which is an approximation of the postcrior probabilitics.
'Thusc confidence valiics may be used for rejecting a test
pattern in cast. of doubt. The radial basis function (about a
Gamsiaii kerncl) is better suited than the sigmoid transfer
function for handling outliers. A radial basis network,
howcvcr, is usually traiiicd differently than a multilayer
porccptron. Instead o f R gradient search on the weights,
hidden neurons arc added unti I some preset performance is
rcnchcd. The classification restilt is comparable to sitiiations
whcrc each class cnnditional density is represented by a
wcighted slim cif Gaussians (a so-callcd Gaiissian mixturc;
scc Scction 8.2).
A special type [if classifier is the dccisirm tree [22],1301,
[129], which is trained by an iterative st!lcction of individual
fcntures that are most saliciit at each node of the tree. 'llic
criteria for feature sulcction and tree gcncratioii iiicludc tlic
information contcnt, the node purily, or Fisher's criterion.
During classification, just those featmes are undcr consideration that are needed for the test pattern under
consideratinn, so feature selection is implicitly built-in.
The must commonly uscd dccision tree classifiers are binary
in nature and use a single fcature at each node, resulting in
decision boundaries that are parallel tn the feature axes
[149]. Conscqucntly, such decision trees are intriiisically
suboptimal l o r most applications. However, thc main
advantage of the tree classifier, besides i t s spccd, is the
possibility to interpret the decision rule in terms of
individual features. 'l'his makes decision trues attractive
for iriteractivt: iisc by experts. Like neural networks,
decision trccs can be. easily overtrained, which can be
avoided by using a priining stage [63], [106], 13281. Decision
tree classification systems such as CART [22J and C4.5 [129]
arc available iti the public domaiti4 and therefore, often
uscd as il benchmark.
One of the mnst interesting recent developments i n
classifier dcsign is the introduction of tlie supprwt vector
classificr b y Vapiiik [I621 which has also bccii stitdied by
other authors [23], [1441, 11461. It is primarily a two-class
classifier. l l w optimization criterion here is the width of the
margin between tlic classes, i.e., the cinpty area around tlic
decision boundary defined by the distance l o the nearcst
training yntterns. These pnlkrns, called support vectors,
finally delinc thc classification [unction. Their number is
minimized by maximizing thc margin.
Tlic decision function for a two-class problem derived by
the support vcchr classifier can bc written as follows using
a kernel iunction TC(s,,a) o f a new pattern 3: (to be
classified) and a training pattern 5 , .
D(z)=
(tiXiIi(Xi,ilZj +(YO,
!/TI t
(izj
s
where ,i* is the support vector set (a subsct of the training
set), and X i - fl Ihe label of object 5 , . The parnmciors
,Y~2 0 are optimized during training by
4. l ~ p/ /www.e,md
:
.ck/id-xchiw/
+ CCCJ
iniiin~o"'~\~<2~(v
(la)
3
constrained by A j f l ( z j ) 2 1 - E : ; , V X , ~in the training set. h i s
a diagonal matrix containing thc labels A j and the matrix IC
stores the valucs of the kernel function IC(xi,x ) for all pairs
of training yattcms. The set of slack variables ~j allow for
class overlap, contrtdlcd by the penalty weight C > 0. For
(7 = CO, no overlap is allowed. Equation (1.3) is thc dual
form of maximizing the margin (phis the penalty term).
During optimization, the vnlucs of all cri become 0, cxccpt
for l:hc support vectors. Sn the support vectors are the only
ones that arc finally. needed. l'he ad hoc character of tlw
penalty term (error penal~y)nnd the computational complcxity of the training procedurc (a quadratic minimization
problem) arc thc Jrawbacks of lhis method. Various
training algorithrns have been proposed in the litcrature
[ 231, including chuiiking [ 1611, Osuna's decomposition
method [119], and sequential minimnl optimiziltion [ 1241.
An appropriate kcrncl fiunction TI (as i n kerncl PCA, Section
4.1) needs to be sclcctccd. In its most simple form, it is just a
dot product between tho input pattern z and a member of
tlic support set: xi xi,^) : x i . x, resulting it) il linear
classifier. Nonlinear kernels, s~ichas
K ( X i , X ) = (Xi ' X
+ l)!'!
result in a pth-order polynomial classifier. Ihussiaii radial
basis functions can also be used. The important advantage
of thc support vector classifier is that it offers EI possibility to
train gencraliznble, iionliriea r classifiers in high-dimcnsional spacw using a s m a l l training sei. Moreover, for large
training sets, it lypically selects a small support set which is
necessary for designing tlie classifier, thcrcby iniiiirniziiig
the camputalitiid requirements during ksting.
'llic support vector classifier can also be understood in
terms of the Iradilioml teemplate matching techniques. 'I'hc
support vectors replace the prototypes with the main
diffcrence being thal: they characterize the classes by a
decision boundary. Morcovcr, this decision boundary is not
just defincd by theminimuin disl-anccf~inction,butby amore
general, possibly non1i near, combination of these dishices.
Wc summarize the most commonly used classificrs in
Table 6. Many of them rcprcscnt, in fact, an entirc family of
classifiers and allow th:hc user to inoclify several associated
paramctcrs and criterion hinctions. All (or almost all) of
these classifiers are adruissible, i n thc sense that there cxist
some classificatinn problems Cor which they are the best
choicc. An extensive cornpmison of a large set of classifiers
ovcr many different problcms is the Statlog prtjcct [I091
which showed a large variability over their relative
performances, illustrating that there is no such hing as a n
nvcrall optimal classificatioii rule.
The differences between the decision boundaries obtained
by different classifiers arc ilhistrated in Fig. 7 using dataset 1
(2-dimensiona1, two-class problem with Gaussian derisi tics).
Note the two small isolated arcas for IT$ in Pig. 7c for the
1-NN rule. The neural network classifier in Fig. 7d even
shows a "ghost" region that seemingly has nothing 10 do
with the data. Such rcgions are less probable for a sinall
numbcr of hidden lnycrs at the cost of poorer class
scpai'a tion.
IEEE TRANSACTIONS ON PATTERN ANALYSlS AND MACHINE INELLIGENCE. VOL. 22. NO. 1 , JANUARY 2000
20
TABLE 6
Classification Methods
A larger hidden layer may result in overtraining. This is
illustrated in Fig. 8 for a network with 10 neurons in the
hidden layer. During training, the test set error and the
training set error are initially almost equal, but after a
certain point (three epochs') the test set error starts to
increase while the training error keeps on decreasing. The
final classifier after 50 cpochs has clcarly adapted to the
noise in the dataset: it tries to separate isolatcd pattcms in a
way that does not contribute to its generalization ability.
6
CLASSIFIER
COMBINATION
Tncrc arc sevcral reasons for cninbining multiple classifiers
to solve a given classification problem. Some of them are
listed bcluw:
1.
A designcr may haw acccss to a number nf different
classifiers, each developed in H different context and
5. Onr cpndi mcnns gniiig tlirough thr! ciitilr! training data uiice.
for an entirely different represcnt~ti~ii/desc1.iption
of tlw some prublcm. An cxainplc is tlic idcntificati.on of persons by their wicc, face, as well as
handwriting.
Sometimes more than a singlc training sct is
available, each collected a t a different time or in a
different environment. These training sets may even
USP different features.
Diffcrmt classifiers trained on thr! samc. data iiiny
not only differ in their global pcrformanccs, but they
also may show strong lociil differences. Each
classifier mny h a w its own rcgicm in the fcatiirc
space where i t performs the best.
Some classifiers such as neural networks show
different rcsults with diffcreiit initializations due to
the randomness inherent in the training procedure.
Instead of selecting the best network and discarding
the others, one can combine various networks,
JAIN ET AL.: STATISTICAL PATTERN RECOGNITION:A REVIEW
21
..-..
- .-
*
*
I
0
Fig. 7. Decision boundaries for two bivariate Gaussian distributed classes, using 30 patlerns per class. The following classifiers are used: (a) Bayesnormal-quadratic, (b) Bayss-normal-iinear,(c)I-",
and (d) ANN-5 (a feed-fonvard neural network with one hidden layer containing 5 neurons). The
regions RI and Jf,i for classes w , and 02, respectively, are found by classifying all the points in the two-dimensional feature space.
thereby taking advantagc. of all the attempts to learn
from the data.
In summary, we may have different feature sets,
different training sets, different classification methods or
different training sessions, all resulting in a set of classifiers
whose outputs may be combined, with thc hope of
improving thc overall classification accuracy. If this set of
classifiers is fixed, the problem focuses on thc combination
function. It is also possible io use a f i x e d combinrr and
optimize thc sct of input classifiers, see Section 6.1.
A large number of combination schemes have been
proposed in the Iiteraturc [172]. A typical combination
schcmc consists of a set of individual classifiers and a
combiner which combines thc rasults of the individual
classifiers to make the final decision. When the individual
classificrs should be invoked or how they should interact
with each other is determined by the architecture of the
combination scheme. Thus, various combination schemes
may differ from each other in their architccturcs, thc
characteristics of the cnmbincr, and selection of the
individual classifiers.
Various schemes for combining multiple classificrs can
be grouped into three main categories according to their
architecture: 1) parallel, 2) cascading (or scrial combination), and 3) hierarchical (tree-like). In the parallel architecture, all the individual classifiers are invoked
independently, and their rcsults arc thcn combined by a
combiner. Most combination schemes in the literature
belong to this category. rn the gakd paralld variant, the
outputs of individual classifiers arc sclcctud or weighted by
a gating device bcforc they are combined. In the cascading
architect~ire,itidividual classifiers are invoked in a linear
sequence. The numbcr o f pssiblc classes for a given pattern
is gradually rvduccd as more classifiers in the scqucnce
have been invoked. For the sake of efficiency, inaccurate but
cheap classifiers (low cnmputational and nieasuremcnt
demands) are considcrcd first, followed by more accuratc
and expensive classifiers. In the hierarchical architecture,
individual classifiers arc combined into a structure, which
is similar to that of a decision tree classifier. The tree nodes,
howovcr, inay now be associated with complex classifiers
demanding a large number of fcalurcs. Thc advantage of
this architccturc is the high efficiency and flcxibility in
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL, 22, NO. 1 , JANUARY 2000
22
I
I
;
. . .__
.....................
.....................................................
I . . . trainlng set error
I
~
:
“‘
I..::.
.::..:...!es!
1
: n
0.2 -
q
error
-.-~
....
~~
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. \............,
>.................................
,i
..I
0
..........................
10
L
....................................
20
30
Number of Training Epochs
I
40
....................
50
Fig, 8. Classification error of a neural network classifier using 10 hidden
units trained by the Levenberg-Marquardt rule lor 50 epochs from two
classes with 30 patterns each (Dataset 1). Test set error is based on an
independent set of 1,000 patterns.
exploiting thP discriminant power of different types o f
fcaturcs. Using these three basic archilccturcs, we can build
even more complicated classifier combination systems.
6.1 Selection and Training of lndlvldual Classifiers
A classificr combination is especially useful if the individual classifiers arc largely independent, I f this is not already
guaranteed by the usc of different training sets, various
rcsampling techniques like rotation and bootstrapping may
be used t r , artificially create such differences. Examples are
stacking [3.68], bagging [21], and boosting (Or ARCing)
[142]. In stacking, the outputs of the individual classifiers
are used to train the “stacked” classifier. The final dccisim
is inade based on the outputs OF thc stacked classifier in
conjunction with the outputs of individual classifiers.
In bagging, different datasets arc created by bootstrapped versions of the original datasck and combined
using a fixed rulc like averaging. 13oosting [52] is aiiothcr
resampliiig tcchnique for generating a sequencc of training
data sets. The distribution of a particular training set in the
sequence is overrepresented by patterns which were
misclassified by the earlier classifiers in the sequence. Tn
boosting, fhc individual classifiers are trained hicrarchically
to learn to discriininatc more complex regions in the feature
space. The original algorithm was proposed by Schapire
[142],who showed that, in principlc, it is possible for a
combination of weak classifiers (whwx pcrformances are
only slightly b c k r khan random guessing) to achieve an
error rate which is arbitrarily small on the training data.
Sometimes cluster analysis may be used to separate the
individual classes in tho training set into subclasses.
Consequently, simpler classifiers (e.g., linear) may be used
and combined later to generate, for instance, a picccwisc
linear r u s d 11201.
Instead of building different classifiers on diffcrcnt scts
nf training pathiis, different feature sets may be used. This
even more explicitly furccs thc individual classifiers to
contain independent information. An example is the
random subspacc method [75].
6.2 Combiner
Aftcr individual classifiers have been selected, they need lo
be combined together by a mudulc, called tlw combiner.
Various combiners can bc! distinguished from each other in
their trainability, adaptivity, a i d rcquircmcnt on the output
of individual classifiers. Combiners, such as voting, averaging (or sum), and Borda count [74] are static, with 110
training required, whilc others are trainable. The trainable
combiuers may load to a better improvement than static
combiners at the cost o f additional training as well as the
requirement of additional training data.
Some combination schcmes arc adaptive in the sense that
the combiner cvduatcs (or weighs) the decisions of
individual classificrs dcpcnding on the input pattern. In
cnntrmt, nonadaptive combiners treat all the inpiit patterns
the saiiic. Adaptive combination schemes can furthcr
exploit the detailed error characteristics and expertise of
individual classifiers. Examples of adaptive combiners
include adaptive weighting [l.56], associative switch,
mixture of local experts (MLB) [79], and hierarchical
MLE [87].
Different combiners expect different types flf output
from individual classifiers. X u et al. [I721 gruupcd these
expectations into three levels: 1) measureincnt (or confidence), 2) rank, and 3) abstract. At the confidcncc level, a
classifier outputs a numerical valuc for cach class indicating
the belief or probability that thc given input pattern belongs
to that class. At the rank level, a classifier assigns a rank to
each class with the highest rank buing the first choice. Rank
value cannot be used in isdatimi bccause the highest rank
does not necessarily mean a high confidence in the
classification. At tho abstract level, a classifier only outputs
a unique class labcl or several class labels (in which case,
the classes are equally grmd). The confidence level conveys
the richest information, while thc abstract level contains the
least amount of information about thu decision being made.
Table 7 lists a number of representative combination
schemes and their characteristics. This is by no means an
exhaustivc list.
6.3 Theoretical Analysis 01 Combination Schemes
A large number of oxpcriinental studies have shown that
classifier combination can improve the recognition
accuracy. However, there exist only a few thcorclical
cxplanations for these experimental results. Morcover, most
cxpIanatioiis apply to only the simplest cnmbination
schemes under rather restrictive assumpticms. One of the
most rigorous theories on classifier combination is
presented by Kleinberg [Sl].
A popular analysis nf combination schemes is based on
the well-known bias-variance dilemma 1641, 1931. Kegressiori or classification ci’ror can be decomposed into a bias
tcrm and a variance term. Unstable classifiers nr classifiers
with a high complexity (or capacity), such as dccisinn trees,
iwircs t neighbor classifiers, and large-sizc ncural networks,
can have universally low bias, but a large variance. On the
other hand, stable classificrs ni- classifiers with a low
capacity can h a w n low variance but a large bias.
Turner and Ghush [158] provided a quantitative analysis
of the improvements in classification accuracy by combining multiple neural networks. They showed that combining
JAIN ET AL.: STATISTICAL PATTERN RECOGNITION: A REVIEW
23
TABLE 7
Classifier Combination Schemes
I
'-
7
- I
I
-1
.
nc>tworks using a liiicar coinbiner or order statistics
combiner redtices thc variance of thc actual decision
boundaries around t1i.h~op tiniunz boundary. In the absence
ofmetwork bias, Lhc reduction i n the added error (to Bayes
error) is directly proportional to the reduclion in the
variance. A linear combination of 11' unbiased neural
networks with indcpendent and idcntically distributed
(i.i.d.) crroi distributions can reduce the variance by a
h c k x uf N . At a first glance, this result suuiids remarkablc
for as N approaches infinity, the variaiicc is redwed to zt'rc).
Unfortunately, khis is not realistic bccatise the i.i.d. assumption brmks do\w for large N. Similarly, Perrane and
Cooper 11231 showed that under tho zero-mean and
independence assumption on the misfit (diffeerencc.between
the dcsircd output a n d the actual output), averaging the
outputs of
neural networks can rcducc the mean squarc
error (MSE) by n factor of N comprlred to the avcragcd MSE
ol: thc N neural networks. For a largc N , the MSE of the!
enscmble can, i n principle, be made arbitrarily small.
Unfortiinately, as mentioned above, the independuiicc
assumption brcaks down as M increases. Pcrrone and
Cooper [123] also proposed a gcncralized enscmblc, an
optimal linear combiner in the least sqiiilrc ci'ror sense. In
the generalized enscmblc, weights are derived from thc
errnr corrclation matrix of the N neural networks. It was
shown that the MSE of thc generalized cnscmble i s smaller
than the MSE of the best neural network in thc ciiscnible.
This rcsult is based on the assumptions that thc rows and
columns of tho crror correlation matrix are linearly
~
independent atid tho error correlation matrix can be reliably
cstimated. Again, tliccsc assumptions break down as M
incrcases.
Kittler et al. [90] dcveloped a c o m ~ n o n theoretical
framework for a class of combination schemes whcrc
individual classifiers use distinct fcaturcs to estiinatc tlic
pnstcrior probabilities given the input pattern. They
introduced a sensitivity analysis to explain why the sum
(or average) rulc outperforms the other rules for the same
class. T h y showed that the s u m rulc is less sensitivc than
titliers (such as the "prciduct" rule) to the error of individual
classifiers in estiinating posterior probabilities. The s u m
ride is most appropriate for combining diffcrent estimates
of the samc posterior probabilitics, e.g., resulting from
different classifier initializatiuns (case (4) in the inhductinn
of this section). The product rule is most appropriate for
combining preferably erl-or-frco indcpeiident probabilities,
e.g. resulting from wcll estimated densities of different,
indupcndent feiltiire sels (caw (2) in the introduction of this
scction).
Schapire el. al. [I431 proposed a diffcreizt exp1analioii for
tlw effectiveness nf voting (weighted avcrage, in fact)
methods. 'Thc cxplanation is bascd on the r-iotion of
"margin" wliicli is the difference between thc combiricd
score of the correcl class and the highcst combined score
among all thc iiicorrect classes. They established khat the
generalization error is bounded by the tail probability of the
margin distribution on training data plus a term which is a
function ol thc complexity of a single classifier rather than
24
IEEE TRANSACTIONS ON PATERN ANALYSIS AN0 MACHINE INTELLIGENCE, VOL. 22, NO. 1, JANUARY 2000
combination ~ L I I C S (last f i w columns). Fur uxamplc, 1he
voting rule (culumn) over the six decision tree classifiers
(row) yields an crrur r)f 21.8 pcrccnt. Again, it is tindertined
10 indicati. that h i s combination result is better than each of
thc six individiial results of the decisioii tree. The 5 x 5
blnck in the bottom right part of Table 8 presents the
coinbinntioii rcsulls, o v c ~ 'tlie six feature sets, for the
classifier combination schemes for each of the separate
6.4 An Example
foaturc sets.
We will illustrate the characteristics of a nuinher of different
Sriinc of tlic classifiers, for example, h e decision tree, do
classifiers a n d combination rulcs on a digit classification not purforni wcll on this data. Also, the neural network
probleni (Dataset 3, see Section 2). 'lhc classifiers used in the clnssificrs providc ratlicr poor optimal solutiotis, probably
experiment were designed using Matlab and wow not duc to noiicoiivcrgjng training sessions. Some of the simple
optiinized for the data set. All the six diffcrcnk fcatiirc s c k clnssilicrs such as tho l-NN, Baycs plug-in, a n d I'arzen give
for the digit dataset disciissed in Section 2 will be iised, good rcsults; the perkmnanccs of d iffei-ent classifiers r7ary
enabling u s to illustrate the performance o f various substantially ovcr diffcrcnt fcaturu sets. Due to the
classifier combining rulw uvw differcnt classifiers as wcll rclativcly small training sck for soinc uf the large feature
as over different feature sets. Confidence values in the sck, tho Baycs-normal-quad I-atic classifier is outperformed
ou.tpir.ts of all the classifiers are computed, either directly by Lhc linear ono, but the SVC-quadratic generally performs
based on the posterior probabilities or on the logistic output better than the SVC-linear. This shows that the SVC
function as discussed i n Section 5. These outputs are also classifier can find nrmlincar solutions without increasing
used to obtain millticlass versions For intrinsically two-class thc ovcr kraining risk.
discriminants such as the Fisher Linear Discriminant and
CunsicIcring-tlic classiiicr comnIination rcsu~ts,it appears
thc Support Vcctor classifier (SVC). For thcsc rwo that the tmined classifier coinbination iulcs arc not always
classifiers, a total of 10 discriminants arc c o m p t c d bclwccn bettcr than the use of fixed rulcs. Still, thc bcst ovcrall icsult
each of the 10 classes and the cornbincd sct cif thc rcinainjng (1.5 percent error) is obtained by a trained combination rule,
classes. A test pattern is classified by sclecting the class Tor the nearest mcan mcthod. Tlic coiiibiiiahi of different
which lhu discriminant has the highest confidence.
classificrs for thc sainc kcatill-c sct (columns in the table)
The following 12 classifiers arc used (also see Table 8): only slightly improves the best individual classification
the Eayes-plug-in rille assuming normal distributions with rcsults. 'I'hc best combination rule for this dataset is voting.
different {Bayes-i~ormal-quadratic)or equal covariance Thc product rulc bchaves poorly, as cc711 be expected,
matrices (nayes-normal-linear), the Nearest Mean (NM) bccausc difforcnt classificrs oil tlic same feature set do not
rule, 1-N.N, LAN, I'arzen, Fisher, a binary decision tree provide indepcndcnt crmfidcnco vnlucs. TIic ct)mbination of
using the niaximum purity criterion [21]and early pruning, results obtained by the same clnssificr over different feature
two feed-forward neural networks (based on thc Matlab sets (rows in the table) frcqucntlp outpcrforms the bcst
Neural Network Toulbox) with a hiddcn laycr coiisisting cif individual classifier restdt. Sometimes, the improveinents
20 (ANN-20) a n d 50 (ANN-SO) neurons arid tlic linear are substantial as i s tlie case for. the decision tree. Here, the
(SVC-liiicar) and quadratic (SVC-quadratic) Supporl:Vcctnr product rule does niucli better, but occasionally it performs
classifiors. Tlic number o f neighbors in tlio k-NN rule and surprisingly bad, similar to the combination of neural
tlic smoothing paramotcr in tlic Parzcn classificr arc both network classifiers. 'L'his combination rule (like the miniop timizcd over tho classificatiun result using tho loavc-ono- mum and maximum rulcs, not uscd in this experiment) is
out error estimate on tlie training set. For combjtiiiig sensitive to poorly trained individual clnssificrs. Finally, it
classifiers, the median, product, iind voting rules are used, is wortlirvhilc to observe that in combining khc ncural
as wcll a s two trained classifiers (NM and :l-'NN). Thc network results, the trained combination rulcs do very well
training set used for the individual classifiers is also used. in (classification errors between 2.1 percent and 5.6 percent) in
classifier com.bii~atioiz.
comparison with thc fixed rulcs (classificatiun errors
The '12 classifiers listed in Table 8 were trained on the between 16.3 percen t to 90 percent).
same 500 (10 x 50) training patterns from each of the six
feature sets and tested on the same 1.,000 (10 x 100) test
patkrns. 'I'he resulting classification errors {in percentage) 7 ERRORESTIMATION
arc reported; for each feature set, tlie best result over the The classification error 01' simply thc i?ri'or ratc, c, is tlw
classifiers is printed in bold. Ncxt, the 12 individual ultimate measure of the performance of ;I classifier.
classifiers for i> single feature set were combiiicd using the Competing classificrs can also bc evaluated based on
five combining rules (median, product, voting, nearest their emir probabilities. Other performance muasuws
mean, and .)1":
For example, the voting rule (row) ovcr iiicludc the cost of mcasuring fcaturcs and thc computabhc classiliicrs using fcatiirc set Number 3 (column) yields tional requirements of tlw dccisioii rrilc. Whilc it is easy
a n error of 3.2 percent. It is underlined to indicate that this to define thha probability d error in terms of thc classcornbination result is better than the performance of conditional dcnsitics, il is vcry difiictilt to obtain a closedindividual classifiers for this feature set. Filially, tlie outputs form cxprcssion for P,. Even in thc rclativcly siniplc cast!
of each classifier and each classifier combination schcmc cif multivariatu Caussian densities with unequal
m7er all the six feature sets are combined using thc fivc crwariaricc malriccs, il is not possible I'o write a simple
the combined classifier. Thcy dcinonstrated that the
bonsting algorithm can effectively improve the margin
distribution. This finding is similar to the property of thc
support vector classifier, which sliows the importance of
tmining patterns near the margin, where the margin is
dcfiiicd as thc area of ovcrlay bctween the class conditional
densities.
JAlN ET AL.: STATfSTlCAL PATTERN RECOGNITION: A REVIEW
25
analytical cxpression for t h c crroi. rate. If an analytical
expression for the error ratc w a s available, it could be
w e d , f o r a given decision r d c , to study tlic behavior of
!<!
as a function of the numbcr of features, true parameter
values nf thu densities, ~ ~ u m b oofr training samples, and
prior class probabilities. Fnr consistent training rules the
value of P,! approaches the Baycs error for increasing
sample sizus. For some families of distributions tight
bounds for khc b y e s error may be obtained 171. For finite
sample sizcs and unknown distributions, howcver, such
bounds arc inipossible [6], [41].
I n practice, the error rate of a recognition system must be
cstimated from all tlic available samples which are split into
training and test sets [70]. The classificr is first designed
itsing training samples, and then it is cvaluated bascd on its
classification pcrformance on tlic k s t saiiiples. Tho percentage of misclossified test samples is taken as an estimate of
the error ratc. In order for this crror estimate to be reliable
in predicting futuri! classification pcrformance, not only
should the training set and the test sct be sufficiently large,
but the training samldes and the tcst samples must be
indcpcndent. This rcquirement of indcpcndent training and
test samples is still oftcn overlooked in practice.
An important point to keep in mind i s that thc error
estimate of a classifier, being a fuiicction of thc specific
training and test scts used, is a random variablc. Given a
classifier, suppose T is the nuinber of test samples (out of
a total of ,n) that are misclassificd. It can be sliown that the
prubability density function of T has il binomial distribution. &'l'hcmaximum-likelihui)d estimate, k!,
of P, is given
by ,'I 7r/n, with E ( = P, nnd
e:(1 Pc)/n.
Thus, I:: is a n unbiased and cnnsistcnt estimator. Because
is a random variable, a confidcncc interval is associated
with it. Supposc 11 = 250 and T = 50 then P, = 0.2 and a
95 percent confidence interval of FE is (O.lS,O.Z5). The
confidetm interval, which shrinks as the numbor 11 of test
ec)
.cp
7
~
samples increases, plays an important role in comparing
two competing classifiers, Cl and C2. Suppose a total uf
100 tcst samples are available and Cl and C, misclassify 10
and 13, respectively, of these samples. 1s classificr GI better
than C2? The 95 perccnt confidence intervals for the true
errnr probabilitics of these classifiers a14c (0.04: 0.16) and
(fl.OO,0.20), rcspectively. Siiice Lhcsc confidence intervals
overlap, w e cannot say that the performance of Cl will
always be superior to that of 4.
This analysis is somewho1
pessimistic duc to positively correlated crroi' estimates
based 011 1hc same test set [137].
Hiow should the availnblc samples be split tu form
training a n d test scts? If the training set is small, then thc
resulting classifier will not be very robust and will have a
low generalization abiliby. On the other hand, if the test sct
is small, then thc confidence in the cstimakcd error rate will
be luw. Various methods that are commonly used to
cstimate the error rate are summarizcd in Table 9. 'I'hcsc
methods differ in how they utilize the available samples as
training and tcst sets. If the numbcr of available samplcs is
extremely large (say, 1. million), tlicn all these methods are
likcly to lead to thc same estimate of the error rate. h r
example, while it is well known that thc rcsttbstitutioi~
method provides a n optimistically biased estimate of thc
error rate, tlie bias becnmcs smaller and smallcr a s the ratio
of the numbcr of training samples per class to tlie
dimensionality of the feature vector gets larger and larger.
Thcre are 110 good guidelines available on how to divicic the
available samplcs into training and test scts; FLIkunaga [58]
provides arguments in favor of using more samples for
testing thc classifier than for designing the classifier. Ni)
maktcr how the data is split into training and test scts, it
should be clear that different raiidoiii splits (with the
specified size of training and test sets) will rceult i n
di ffccrcnterror estimates.
IEEE TRANSACTIONS ON PATrERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 1 , JANUARY 2000
26
TABLE 9
Error Estimation Methods
I .
Fig. 9 shows the classification error of the Bayes plug-in
linear classifier on the digit dataset as a function of the
number of training patterns. Tho test sct error gradually
approaches tlie training set error (resubstitution error) as
the number of training samples increases. The relatively
large difference between these two error rates for 100
training patterns per class indicates that the bias in these
1wo w r w cstiinates can bc furthcr reduced by enlarging thc
training set. Both the curves in this figure represent the
avemge of 50 experiments in which training sets of the
given size are randomly drawn; the test set of 1,UUO patterns
is fixcd.
Thc holdoul, leave-ane-rml, and rotation methods arc
versions of tlic! cross-validation approach. Onc o f thc
main disadvantagcs of cross-vnlidatirm mcthods, cspccially for small snmplc size siimtions, is t11a1 not a11 the
available
samples are used for. training the classifier.
Further, the two extreme cases of cross validation, hold
out mcthod and Icavc-uno-out method, suffer from cithcr
large bias or large variance, respectively. To overcome
this limitation, tho bootstrap method [48] has been
proposed to estiinatc the wrm rate. Thc. brwkstrap mcthod
resamples the available patterns with replacement to
gcmratc a numbcr of ”fake” data sets (typically, several
hundrcd) of th:hc Same size as the given training set. These
new training sets can bc uscd not only tn cstirnatc the
bias of the resubstitution cstimatc, b u t also t u ddiiic
other, so called bootstrap estimates of the error wte.
Experimental results have shown that thc bunktrap
estimates can outperform thc crnss validatinn cskimatcs
and thc resubstitution estimates of the error rate [82].
In many pattern rccognitim applications, it is n o t
adequate to characterize the performance of a classifier by
B singlc niimbcr, P,, which iiicasures Ihc overall error rate
of o systcm. Consider t11c prciblein of evaluating a
fingerprint matching system, where two different yet
rclated ermr ratcs arc of interest. Thu False Acccptance
Itate (E’AIZ) is the ratio of the number of pairs of different
fing~rpi~ints
that arc iiictxrectly matchcd by a given system
to the total number of match attempts. The False Reject Rate
(FRR) is thc ratio of the number of pairs of the same
fingerprint that are not inatcliod by a givcii system to thc
total number of match attempts. A fingerprint matching
system can be tuned (by setting an appropriate threshold on
the matching score) to operate at a desired value of FAR.
However, if we try to decrease the FAR of the system, then
it wciuld increase the F1ZK and vice versa. The Receiver
Operating Characteristic (ROC) Curvc 11071 is a plot uf FAR
versus FRR which permits the system designer to assess the
performance of the recognition system at various operating
points (threshr~ldsin the decision rule). In this sense, KOC
provides a more comprehensive performance measure than,
say, tlw cqual error rate of the system (where FRR = PAL<).
Fig. 10 shows thc ROC curvc for thc digil datasct whcrc thc
Baycs plug-in lincar classifiur is Lrained on 100 patLenis per
class. Examples of the usc of ROC analysis arc coinbining
classifiers 11701 and feature selection [993,
I:n addition to the error rate, another useful performalice measure of a classifier is its reject rate. Suppose a
test pattern falls near the decision boundary between the
two classes. While the decision rule may be able to
correctly classify such a pattcm, this classification will be
JAlN ET AL.: STATISTICAL PATTERN RECOGNITION: A nEVlEW
27
0,
r~
E
2 0.04
test skt c m r
training set arror
.....................................
I
0.09
.
.......,.... ......................................................
, ............
~
.
-
B
0.02
OO
---20
I
40
--60
SO
ol
.........................
1
I
-
0
A .-
0.02
....
I.
.... - .
......................
0.04
.......
1
O.OG
........
..............
................
L.
0.08
0.1
False Rejoct Rato
Fig. 10. The ROC curve of tho Bayes plug-in linear classifier forthe digit
dataset.
made with a low confidcncc. A better alternative would
bc to reject these doubtful patterns instcad of, assigning
them to one of thc categories under consideration. How
do we decidc when to rejecl a test pattern? For thc Bayes
dccisiom rule, a well-known reject option is to reject s
pattern if its maximum a posteriori probability is bclow a
threshold; the larger the threshold, the higher thc reject
rate. Tnvoking the reject option reduces thc wroc rate; the
largcr the reject ratc, thc smaller tlic m o r rate. This
relationship is Icprcsented as an error-reject trade-off
curve which can bc used to set the desired opcraliiig
point of tho classifier. Fig. 11 slimvs the error-rLjcct ctirve
for the digit dataset when a Bayes plug-in linear classifier
is used. This ciirvc is monotonically iioii-increasing, since
rejecting inmc patterns eithcr reduces the Prror rate or
keeps it the same. A good choice for ~ h creject rate is
bnscd 011 tlie costs associated with rvjcct ancl incorrect
decisions (SCC I661 for an applied example nf the use of
error-reject curves).
8 UNSUPERVISED
CLASSIFICATION
In many applications of pattern recognition, it is extremely
difficult or cxpcnsive, or eveii impossible, to reliably label n
training samplc with its true category. Consider, for
Example, tlie applicatim of land-use classification in rcinotc
sensing. In cirdcr to obtain the “ground truth” information
(catcgwy for each pixel) in tlic image, either the specific site
associated with the pixel. should bc visited or its catcgory
should be extracted from a Geographicnl Information
System, i f onc is available. Unsupcrvised classification
rcfcrs to situations whcrc the objeclivr is to construct
decision boundaries based on unlabelcd training data.
Unsupervised classification is also known as data clustering
which is a generic label for a varicty of procedures designed
to find natural groupings, or clusters, in multidimciisional
data, based on rncasurcd or perceivcd similarities m ” g
the patterns 1811. Category labels and other information
about the sotirce of khc data influence the interpretation of
the clustcring, not the fnrmation of the clusters.
Unsupervised classification or clustering is a w r y
diffkiilt problcm because data can rcvcal cltisters with
different shapes and sizes {see Fig. l2). ‘1-0 compound the
prublein further, the number of clusters in the data oftcn
depends on the rcsoliition {finsvs. coacsc) with which we
view the data. Ono cxample of clustering is the detection
and ddiiioation of a region coiltailling ii high dcnsity of
patterns compared to the background. A numbcr of
hmctiolzal definitions of a cluster have bocn proposcd
which include: 1) patterns within a cluster are mom similar
to each other than are pattcrns belonging to different
clusters and 2) a cluster consisls of a relatively high dcnsity
of points separakl fi.om other clusters by n relatively low
dcnsity of p i n t s . Bvcn with these functional dcfinitions rrf a
cluster, it is not easy to cmnc up with an operational
definition of clustcrs. 01ieof the challciigcs is to seloct ill1
appropriate measure r)f similnrity to define cluskcrs wl.~ich,
in gciicral, is both data (cluster shape) and context
dcpendent.
Cluster analysis is a very important ancl useful tcchnique. ?‘he spocd, reliability, and ccmsistcncy with wliich a
clustcring algorithm c m organize large amounts of data
uonstitutc uverwhelming rcasons to use it in applications
such as data mining [MI,inkorinntion retrieval [17], [25],
image segmentatinn 1551, signal coinprwsion and coding
[‘l],
and iiinchiiie learning [25].
As a conseyenco, huiidreds
of clustering algorithms h a w bccn proposed i n tlie
litcrature a n d iicw cliistering algorithms continue to
appear. However, m n s l of these algorithms arc based rm
thc following two pupiilar clustering techniques: itei-ativc
square-ctror partitioinal clustcring and agglomerative hierarchical clusl-c.ring.Hierarchical (cchniques organize data in
a nestcd scqiieiice of groups which can be displayed in the
form of a dendrogram or a trcc. Square-error partitional
algorithms attempt to ubhiiii that partilion which minimixes
the within-clus tcr scatter or maxiinizcs the betwetin-cluster
scatter. To guarantee that a n optimum solution has been
obtained, one has to cxaininc all possible partitions nf tlic n.
IEEE TRANSACTIONS ON PAUERN ANALYSIS AN0 MACHINE INTELLIGENCE, VOL. 22,
28
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1.. . . . . . .
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0
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0.1
i
0.2
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Fig. 11. Error-reject curve of the Bayes plug-in linear classifier for the
digit dataset.
NO. 1, JANUARY 2000
Fig. 12. Clusters with different shapes and sizes.
algoritlim €or discrctc-valued data. The technique of
coiiceptiiill clustering or learning from cxamplcs [ 11)8] can
bc uscd with pattcrns rcprcscntcd by nonnumeric or
symbolic descriptors. Thu objeclive here is to group patterns
intn cnnceptually simple classes. Concepts are defined in
terms of attributes and patterns are arranged into a
hi~rarcliyof classes described by concepts.
In the following subsections, we briefly summarize the
two most popular approaches to piirtitioiial clustering:
square-error clustering arid mixture decomposition. A
square-error clustering method can bc viewed as a
particular case of mixture decomposition. We should also
point out the differencebetween a clustering criterion and a
clustering algorithm. A clustering algorithm is a particular
implcincnhtion of a clustcring criterion. In this sense, there
are a large number of square-error clustering algorithms,
each minimizing thc squnrc-error criterion and differing
from the others in the choice of the algorithmic parameters.
Some of the welL-known clustering algorithms are listed in
Table .l.O [81].
d-dimensional patterns into Ji' clusters (for a given IT),
which is not computationally feasible. So, various heuristics
arc uscd to rcducc thc scarch, but then t h r c is no guarantci.
of optimality.
Partitionill clustering techniques are used more freqiiently than hierarchical techniques in pattern recognition
applications, so we will restrict our coverage to partitional
mcthods. Rccent studics in cluster analysis suggcst that a
user of A cluskring algorithm should kcep the following
issues in mind: 1) every clustering algorithm will find
cliisters in a givcn datasct wholhur they cxist or not; the
data should, therefore, be subjected to tests for clustering
tendency before applying a clustering algorithm, followed
by a validation of the clusters generated by the algciriitim; 2)
there is no "best" clustering algorithm. Therefore, a user is
advised to try scvcral clustcring algorithms o n R 8.1 Square-Error Clustering
givcn dataact. Further, issuus of data ctrllcclion, data The most commonly uscd partitionnl cluslering s tratcgy is
reprcsmitation, normalization, and cluster validity are as based on the square-error criterion. Tlw gciicral objcctivc
is to obtain that partition which, l o r a fixed number o f
important as the choice of clustering strategy.
The problem of partitional clustering can be formally clusters, minimizes thc squarc-m"'. Suppose that the
stated as follows: Given n patterns in a d-dimensional given set of TL patterns in d dimcnsions has somehow
mctric space, determine a partition of the patterns into fc bwn partitiuncd into K clusters {C',~C ' 2 , ' ' , C:}such Lhat
clusters, such that the patterns in a cluster are more similar clustcr CAhas 'ILL patterns and cach pattern is in cxactly
'q:-: n..
to each other than to pattcrns in different clusters [MI. Thc m e cluster, so that
The mean vcctor, or ccntcr, r)f clustcr Cl, is defined as tlic
value of K may nr may not be specified. A clustering
criterion, either global or local, must be adopted. A g l o h l centroid of tlie clustcr, or
criterion, such as squwe-error, represents each cluster by a
prototypc and assigns the patterns to clusters according to
(Id)
the most similar prototypcs. h lucd critcrion forins clusters
by utitizing local structure in the data. For example, clusters
where x!" is the ith pattern belonging to cluster Ck. The
c m be formed by identifying high-dcnsily regions in thc
square-error For cluster L', is tlie sum of the squared
pattcm spacc or by assigning a pattern and its k nearest
Euclidcan distances between each pattern i n CA and its
neighbors to thc same cluster.
cluster ccnter m(k).This squarc-wror is also cnllcd tlic
Most of tlie partitioiial clustering techniques implicitly
within-cl irstei- variation
assume con tinuous-valued fcahirc vectors so that the
patterns can be viewed as being embedded in a metric
space. If the features are on a nominal or ordinal scale,
Euclidean distances and cluster ccntcrs arc not very
meaningful, so hicrarchical clustering methods are nor- Thc squarc-crror for thc cntirc cluskcring containing IC
mally applied. Wong and Wang [169] proposed a clustering clusters is tlic sum of thc within-clustcr variations:
EL,
JAIN ET A L . STATISTICAL PATTERN HECOGNITION: A REVIEW
29
TABLE 10
Clustering Algorithms
---
. ._
K -mt?anR
Mutual Nciihorrhritd
SinjjeLink (5L)
The objcctive of a square-crrur clustcriiig inethod is tn find n
p a i ~ i t i o i icotihiniiig .IC clristcrs that miiiiriiizcs
for a
fixcd IC. The resulting partition has also bccn referred k o a s
the miiiitiium variance partition. A general algorithm for
the iteralive partitional cluslcriiig inethod is givcn below.
Agori Ihm foi. iteralivc pnrtitiorlal c Iuskring:
Stcp I. Sclect an initial piirtition w i h I< clusters. Kcpcat
stcps 2 throiigh 5 unlil tlic cluster mciiibcrship stabilizes.
Slcp 2. Generak a ncw partition by nssigiiing each pattwn
to its closest cluster center.
Stcp 3 , Crnnputc new rlustcr ccntcrs as the ccntroicis of the
clusters.
Slcp 4. Rcpcat steps 2 and 3 until a i l optimum value of:thc
criterion f u n c h i is found.
S k i ? 5 , Adjust tho nuinbcr of clustc!i's by mei:ging and
splitting cxisting clusters or by removitig small, 01'
oiitlicr, 'clustrrs.
The above algodlm, without step 5, is alst) known as thc
K-means algorittim. 'rhu dctails of the steps in this algorithm
must eitlicr bc supplied by the USCT as parameters or be
implicitly hidden in thc computer propmi. [ [orvcver', thcsc
dctails arc crucial to thc S U C C ~ S S of tlw program. A big
frustration in using clustering progranis is the lack of
pii d el iiws avnilablc for choos i ng A",
initial partition, updating
the parlilioii, adjusting the nurnbcr of clusters, and the
stopping cribxion [&I.
Thc simple IC-tncniis partitioiial dustwing algorikhm
describcd abovc i s coinpiitationally efficient and gives
surprisingly good resul~s if tlw clusters a r c compack,
hypcrsphcrical i n shape and well-separated iri the featiirc
space. If the Mahalanobis distance i s used in defjniiig ilic
sqiiarcd crror in (lh), tlwn the algoritlim is wen ablc to
detect hyporcllipsoidal shapod clusters. Numerous
iitienipis h a w bccn made to impriivo the performance of
the basic IC-means a l g o r i h i i by 1) incorporating a fuzzy
criterinn function lis], resulting in a fuzzy ~ - i n e a n s(LIT
r:-mcans) algorithm, 2) using genetic algorithms, simiilakcd
armding, deterininiskic annealing, and tabu scarch to
30
IEEE TRANSACTIONS ON P A T E R N ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 1. JANUARY 2000
optimize the resulting partition [llO], [l39],and 3) mapping
it onto a neural nulwork 1'1031 for possibly cfficicnt
implementation. tlowcvor, m a n y o f tliosc so-called
enhancements to the K-moans algorithm arc' computationally demanding and q u i r e additional uscr-spccilicd
parameters for which no general guidclincs arc nvailablc.
Jtxdd et al. 11181 show that a combination of algorithmic
enhanceinents +o a sqtiarc-c!rror clristcring algrwithni and
distribution nf the computations Over a tictwotk o f
workstations can be used to cluster hundreds of thousands
of multidimensional patterns in just rl fcw mitiutw.
It is ii-ikc'rcsting. bo nokc how sccmingiy diifcrcnt concepts
for partitioiial clustering can lcad ti, cssc?ntially bhc s m "
algorithm. It is easy to wrify that Lhc gcncra1izc.d Elnyd
vcctr~rquanhzation algorithm uscd in thc commutiicaiion
and compression domain is equivalent to the K m e a n s
algorithm. A vcclur qiiantizcr (VQ) is duscribcd as a
cr,mbinakirm of an cncodcr and a dccodcr. A tl-dimensional
VQ consists of two mappings: a n encoder ,: which maps the
input alphabet (A) to the channel symbol set (M), and a
dccndcr (9 which maps klic chanric!l symbol sc?k (Mj ti, Lhc
output aIpIiabet (A), i.e., +j : A M and p(v) ; kr + A .
A distortion measure 'P(slyj) specifies the cost associ<itrd
with quantization, where 6 = $ ( : ( ( : ( j ) ) , Usually, a11 optiiniil
quantizer minimixs thc averagc distortion mdcr a size
constraint on M. Thus, the problem of vector qiiantization
can be posed as a clustering problem, where the nuiiiber. of
cjustcrs I< is nnw the sizc of the output alphabet,
A : {Iji,
i -- 1 , . . . K ) , and the goal is fr, lind n quaiilization
(referred to as a partition in thc IC-mcans a1gr)rithm) of tlic
&dimensional fcahirc space which miniiniscs thc average
distortion (mean squarc crror) of tlic input pattcms. Vcctor
quantization has b w n widely used in ii muiiibcr r)f
coinprcssicin aiid coding applications, such as spwcti
wavcforni coding, iinngc coding, etc., wliorc o n l y khc
symbols h r thc output alphabet o r the cluster centers are
transmittcd inskcad o f thc enlire signal 1671, 1321. Vcctor
quantization also p v i d e s an cfficicnt tuul for density
estimation [68]. A kernel-based approach (e.g., R mixture of
C;aussisii kernels, where each kernel i s placed at a cliister
ccntcr) can be used to cstiinatc thc probability dcnsity of thr
training snmplcs. A major issuc in VQ is the sclcction oi thcl
output alphabet size. A number of techniques, siich as the
minimum dcscription lcng th (MDL) principle 113x1, can be
used to sclect this paramcltur (see Section 8.2). The
supervised vcrsion o f VQ is called Icarning vricLor quantization (LVQ) 1921.
-
8.2 Mixture Decomposition
Finite mixtures a r e a flexible aiid powerful probabilistic
modcling tool. In statistical paltcrn recognition, the m a i n
use 01 mixtures is in defining formal (i.e., model-based)
ayproaclics to unsuporviscd classilicatirm [HI. '1'Iip reason
bclund this is that mixlurcs adequately model situations
where each pattern ha5 bccn prtiductld by onc (11' a set of
alternative (prolsabilistically modolcd) sourccs (1%I. Ncvcrtheless, it should be kept in iniitd that strict adhcrcncc t u
this interpretation is not reqiiircd: iiiixtiircs caii also bc sccn
as a class of models that are able to reprcscnt arbitrarily
complex probability dcnsity funclions. This makcs mixtures
also well suitcd for rcprrwnting cumplex class-condit ional
densities in supwviscd leartiing scenarios (see [I371 and
refcrenccs thcrein). Finite! niixturcs caii also be used as a.
feature selection tool [ 1271.
8.2.I Basic Definitions
Consider the following scheme for gciicraling random
samples. There are Ti random soiirccs, cach charactcrizcd
by a probability (mass or density} fitnction p,,~(ylO.zr,),
paramctcrizcd by 0",,, Cor rii. - 1 , ..., ii. Each time a sample
is to bc gcncratcd, W P raiidrmily choose one of these
soiii'ccs, with probabilities (n13...!n.),], and thcn sample
from the clioseii sotme. Thc riinclom varinblc dcfiricd by
this (twn-stage) coimpouiicl generating mochnnism is chnractcrizcd by a finitr? mixture distribution; formally, its
p b a b i l i t y function is
I
ir,-
whcrc c;icli p,),(ylO(,))is callcd a coinponetit, and
@)i,rj = { U , - , . . > U l < ! < k I :...!R!<. , } . It is iisunlly assuiiicd thal
all the components liave the samu functional form; for
example, they i1l.e all multivariate Gaussian. Fitting a
mixture model to a set of observations y :
{y(1j:...qy(72)}
consists nf cstimating the set of iiiixture parameters that
best describe this d a t a . Althougli mixtures C ~ U Ibe built from
many ciiffcreiit typcs of coniponiwts, h e majority oC the
lilcralurc ~ocust~s
on (;aiissian 111ixtii res [:I551.
Thc two fuudamcnhl issui.s arising i n mixture fitting
are: 3 . ) how to cstiimtc the parnmctcrs defining khc iuixture
model and 2) how to cxtimatc thc iiumbor o f crnnponunts
[ I N ] . For thc Cimt question, the staitdiird answer is the
q r : c l n I io 17 - m x iir 1 izo tion (EM) algorithm (which, 1111der mild
conditions, converges to the maximum likelihood (MI.,)
cstimatc of tht! mixturc paramclers); s c w r a l authors have
also ad\wcatcd the (coiiipuiaiioiially demanding) Mnrkov
chriiii Moritc-Cirrlo (MCMCI) method [I 351, The aucond
rpostion is iziorc. difiicult; w v c r a l tcchniqiies have been
proposed whicli are surnmarizud in Section 8.2.3. N o k (lint
the nutput nf tliP iiiixtiire decoiiiposition is 3s good as tlic
vnlidity of thc assumud component distributions.
8.2.2 EM Algorithm
Thc cxpcctatioii-inaximiztltioti algorithm interprets the
given observations y as iircourrrpicfr clatn, with the missing
part being a set of labcls associated w i h y ,
jl; r
{x"; ....dh)},
!
Missing varinblc zc') -- :z\'
$.)I"'
indicates wliicli of ttie rc
compoiicnls gcncrntml ?/ I it was the ./rlth coniponent,
then ZIP = I and 2;:) = O,for p ' i n (1551. 111 t~icprcscncc of
both y nncI A, thc (complete) log-likelihood can be writtenas
+
(18)
Thc EM algori thtn proccwls by alternatively applying the
following two skcps:
JAlN ET AL.: STATISTICAL PATERN RECOGNITION:A REVIEW
E-stcp: Compute thc conditional cxycctation of the
complete lag-likcliho_od (given y and the currcnt
parxvetet cstiinate,
Sincc (la) is linear in the
missing variables, tlie &-step h r mixtures reduces to
the computalion of the conditional expectaiitm of the
missing variables: if)!;:) ~ [ ~ ~ j y],
@&.
MI-step:Update the parameter estimatcs:
@(yo).
For tho mixing probabilities, this bccriincs
‘In thc Gmssian c a w , cadi U,,, consists of a mean
vector and a. covariance matrix which arc updated
using weightcd versions (with weights S!::’)) uf the
standard ML estimates 11551.
The main difficulties in using EM for mixture model
fitting, which are current research topics, are: its local
nature, which makes it critically dependent on initialization;
the possibility of convergence to a point on the boundary of
the parameter spacc with unbounded likelihood (i.e., onc of
the e,,,
approaches zcro with the corresponding covarianct!
becoming nrbitia ril y clrw to siizgula r ) .
31
sample from the full rl pusturiori distribution where IC is
included as a n unknown. Despite their formal appcal, we
think Lhat MCMC-based tcchniqucs ~ T Fstill far too cnmputationally demanding to be useful in pattcrn recognition
applications.
Fig. 13 shows an example of mixture decomposition,
where IC is selected using a modified MDL critcrion [SI].
Thc data consists cif 800 two-dimensional pattcriis distributed Over throc Gaussian componcnts; two of the components have the samc mean vector but different covariance
matrices and that is why one dense cloud of points is inside
another cloud of rather sparse points. The level curve
contours (of constant Mahalanobir distance) for thc true
underlying mixture and the estimatcd mixture are superimposed an the data. For details, see [Sl]. Nok that a
clustcriiig algorithm such as IC-means will not be able to
idenlify these three componcnts, due to the substantial
ovcrlap of two nf thcsc components.
9 DISCUSSION
In its early stage of dcvcloyment, statistical pittern recognition focused mainly on the core of the disciplinc: The
Bayesian decision rule and its variclus derivatives (such as
liiieai: and quadratic discriminant functicms),density estimntion, the cursc cif dimensionality problem, aiid e r r w
estimation. h e to the limited computing powcr awilablv
8.2.3 Estimating the Number of Components
in thc 1960s and 1370s, skatistical pattern rccognition
The MI, criterion can izot be used to cstimate thc number of employed rclalivcly simple t e c h iqws which were applied
mixlure ccliiiponents bccausc the maxirnizcd likelihood is a
t n small-scale problems.
imtidecrcnsing function of K , thereby making it useless as a
Since the early 1‘3ROs, statistical yattern rcctipition has
model selection criterion (selecting a value for IC in this
expericnccd a rapid growth. Its frontiers h a w been
case). This is a particular instance of the idcntfiiabilify expanding in many dircctinns simuItaneously. This rapid
problem where fhc classical (xy-basd)hypothesis tcsting expaiisioii is largely driven by tlie following forces.
cannot bc used because thc iiccessary regularity conditions
are not inct [155[.Several altcriiative apprtiaches that h a w
Increasing intcraction and collaboration anioiig
been proposed arc summarizcd bclow.
different disciplines, including neural networks,
E’M-bawd approaches use tlic (fixed K] EM algorithm to
machine lcariiing, statistics, mathcmatics, cornputer
obtain a sequu~ceof paramctur estimates for R range of
scionco, and biology. ‘I’hcscmdtidiscipli nary efforts
values of A’, { (3[,(, .K = IC,,,!,, _. KI,,j,x};the cstiiiiate of K
have fostered iicw ideas, methodologics, and techis then defined as the minimizer of some cost function,
niques which enrich tlw traditional statistical pattern
recognition paradigm.
The prcvalciice of fast proccssors, thc Internet, large
and inexpcnsive memory and storngc. The advanccd
Most oftciz, this cost function Includcs the maximized logcomputcr teclinology has made it possible to
likelihood function plus an iidditional tcrm 5vlzose rolc is to
implcment complex lcarning, searching and optimipenalize largc valttes of IC. A n obviotts choice in this class is
zation algorithms which was not feasible a few
to use the minimum description longth (MDI-1 criterion [lo]
ducadcs ago. It also allows its to tackle large-scale
[138], but several other model sdcction criteria have been
real world pattcrn recognition problcms which may
proposed: Schwarx’s Bayesian inference critcrioii (]>IC),thu
involve millions of samples in high dimensional
minimum mcssnge length (MML) criterion, and Akaike’s
spaces (thousands of features).
information criterion (AIC) 121, 11481, [167].
Emerging applications, such as data mining and
Resampling-bnscd schemes and cross-validation-type
document taxonomy c reat inn and iii aintcnance.
approaches have also been s u ggestcd; these techniques are
‘L’hesc cmcrging applications have brought new
(somputationally) tnuch closer to stochastic algorithms than
chollcnges that foster il roncrved interest in statistical
to the methods in the previous paragraph. Stochastic
pattern recognition research.
East, but not the least, the need for a principled,
approaches gcncrally involw Markov chain Monte Carlo
(MCMC) 11351 sampling and arc far more computationally
rather than ad hoc approach for succcssfully solving
pattern recognition problems in a predictable way.
intensive than EM. MCMC has bccn used j t i two different
For example, inany concepts in neural nclworks,
ways: to implement model selection criteria to actually
which IWI’C inspired by binlogical iieu ral nctworks,
estimatc IC; and, with a m t m “ful1.y Bayesian flavor”, to
~
~
.i
32
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S
Fig.13 Mixture Decomposition Example
can be directly treated in r\ principled wr\y in
statistical pat tern recognition.
9.1 Frontiers of Pattern Recognition
'I'able 1 1 summari7es several topics which, in mi-opinion,
ale at thc frontiers of pattern rccognition. As wc can sce
from Table 11, many fundamental research problems in
slatistical patkrn recognilion rcrnain at thc lorclront cvcn
NO. 1 . JANUARY 2000
as the field continues to grow. One such cxamplr?, model
selection (which is an important issw in avoiding the curse
of dimcnsionali ty), has bcen a topic of contintiecl research
intcrcst. A common practice in model selection relies 011
cross-validation (rotation incthhod), where the best model is
selected based on the performance on thc validation sct.
Since the validation set is not used in training, this method
does not fully utilize the precious data f o r training which is
especially undesirable when the training daki set is small.
To avoid this prnblem, rl number of model selection
schemes [71] have becn proposed, including Baymian
methods ['14], minimum description leiiggth (MDL) 11381,
Akaike information criterion (AIC) 121 and margiiializcd
likelihood [loll, [159].Various other regularization schemes
which i n c o r p m k prior kiwwlcdgc about model structure
and parameters h a w also bccn proposed. Structural risk
minimization based on the notion of VC dimcnsion has alsr)
bcen wed for model selection where the best model is the
011cwith thc bcst worst-cnsc pcrhjrmance (upper bound on
the generalization errur) [162]. However, these methods do
iiot rcducu khc complcxity of tlic search for the best tnodel.
Typically, the complexity incnsurc has to be cvduntcd for
every possible model or in H set of prcsyccified models.
Ccrtain assumptions ( ~ . g .parameter
,
independence) are
often mndo in order to simplify the compkexity evaluation.
Model selection based on stochns tic complcxity has bccn
applied to feature selection jn both supervised learning and
unsupervised lcariiing I l S U ] and pruning i n decision
TABLE 11
Frontiers of Pattern Recognition
lbpic
........
-
Model selectbn and geowallmtion
.
Emple;l
CoInmeLlts
Haymian leasnlng, M I N A ,A E ,
inargllnalw~dlikelihood, structurd
Make full um of' the uv&blc
far ;ranin#.
. .
data
JAlN ET AL,: STATISTICAL PATTERN RECOGNITION: A REVIEW
trccs [lnh]. In the lattcr caw, the best numbcr cif clusters is
alsr, automatically detcrminud.
Another example is mixture inodeling using EM algorithm (sc‘c Scction %2), which was proposed in 1977 [36],
and which is now a very popular approach for density
cstimation ancl clustcring [1591, clue to thc computing
power available today.
Over thc rcccnt years, ii numbcr o f ncw concepts aiid
t-cchniques have also bcm introduced. For i?xnmple, the
maximum margin objcctive was introduccd in the context
of support vector machines 1231 based on strtictural risk
minimization tlwory 1162j. A clnssificr with a large margin
separating two classes has a small VC dimension, which
yields a gcid generalization performance. Many succcssful
~ipplicationsof SVMs have demonstrated the superiority o f
this objective. function over others [72]. It is found that the
boosting algorithm [I431 also improves the margin distribu tion. The maximum inargin objeclivr! can be considered as a special rcgu1ai:ixed cost function, where the
regularizcr is the inverse of the margin between tho two
classes. Other ri.gularized cost functions, such as weight
decay and weight climjnation, haw also been used in the
context of neural networks.
Due to klic introduction of SVMs, liiwar and qundmtic
programming optimization techniques are once again b h i g
oxteiisivcly studied for pattorn classification. Qundratic
programming is credited frir lcading to the nice propcrty
that tlie decision bnundary is fully specificd by boundary
patterns, while linear programming with thc L‘ izorm or the
invcrse of tlie margin yiolds R small set oE fcaturcs when the
optimal solution i s obtained.
Thc topic of local dccisicm boundary learning has also
received o lot of attention. Tts primary emphasis is on using
patterris ticar the boundary of difkront classes to conslruct
or modify thc dccision boundary. niic such an examplc is
the boosting algorithm and its variatioi’l (Adalloost) tvhcrc
misclassified patterns, mnstly mar the decision bmmdary,
are subsampled with higlicr prohbilitics than correctly
classified patterns t o fnrm a new training sat for trajning
subsequent classifiers. Combination o f local experts i s also
related to this cmccpt, since local experts can learn local
decision boundaries more accurately than global methods.
‘ln gcncral, classifier combinalicm could ref inc dccision
boundary such that its variancc with respect to Baycs
decision briuiidary is reduced, lcading to improved
recognition accurwy [l58].
Scqucntial data arise in many real world problems, such
as speech and on-line handwriting. Sequential pattcrn
recognition has, thccoforc, become a very important topic
in pattern recagnition. Hidden Markov Mndcls (HMM),
have been a popular statistical tool for modeling arid
recognizing seqiiential data, in particular., speech data
[130], [MI. A large number of voriatinlls and enhancemmt~
of HMMs have been proposed in the literature [12],
including hybrids of HMMs and neural networks, iilplitoutput HMMs, weighted transducers, variable-dilration
HMMs, Markov switching models, and switching statcspace models.
Thc growth i n sensor tcchnrhogy and compuling
powor has enriched the availability of data in sevcral
ways. Real world nbjccts can now be rcyresented by
33
inany mure iiieasurcments and satnpled at high ratcs. As
physical objccb h a w il finite complexity, these mcasmcinents are gcncrnlly highly correlnted. This explains why
models usirig spatial and spcctral corrclation in imagcs,
or the Marknv structure in speech, or subspacc approaches in general, liave become so important; tlwy
compress the data to what is physically meaningful,
thereby improving the classification acciiracy simultaiieoti s1.y.
Superviscd learning requires that every training sample
be labeled with iks truc category. Collecting a largc amount
of labeled data can sometimes be very expensivc. 111
practice, we oftcn h a w a small amount of 1abelc.d data
and a larga amount of iinl~~beled
data. ‘I-IOW to makc use of
unlabeled data lor training a classifier is a n important
problem. SVM has bccn cxtcnded to perform scmisupervised learning [3.3].
hivariant pattern rccognition is desirable in many
applications, such as charnctcr and face recognition. Early
research in statistical pattern recognition clid umphasize
cxtraction of invariant h t u r c s which tiiriis otit to bc n vcq(
difficult task. Recently, thcru has been soiiie activity in
designing invariant rccognition inetliods which do not
require invariant fcaturcs. Examples are the noarest
neigltbor classificr using tangent distance [ 1,421 a i d
~
d.eforniable tcmplak matching [MI.These approachus D I ly
achievc invarinncc to small amounts of linear kransformations and nodincar deformations. Besidcs, h e y are
computatirmally very intensive. Simard ct al. [153] pic>posed an algoritlim named. Tangent-1’1-op to minimise the
derivative of tlic classifiei, outputs with rcspcct to distortion
parameters, j ,e,, to improve tlic iiivariance properly rif the
classifier to the selected distorlinii. ‘l’hismakes the tixincd
classificr comptitationally very cfficicnt.
It is well-known that thc human recognition proccss
relics hcavily on context, knowlcdge, and cxpcrience. ‘Tho
cffectiveness of using contextual information in rcsolviiig
anibigLlity and recognihg difficult patterns in thc major
differentiator betwccii rccognition abilities of human beings
aiid machincs. Contextual information has bccn successfully uscd in spccch recognition, OCR, and reiiiotc sensing
I.1731. It is commonly used as a puslproccssiizg step to
corrccl: mistakes in the initial rrcognilion, but there is rl
recent trend to bring contextual infcmnalicm in the earlier
stages (e.g., word scgmcntation) of il recognition system
[174]. Context informiition is often incorporatcd through the
use of compound decision theory dcrivcd from I3aycs
theory iir Markovian models [175]. Chic rcccnt, successful
application is the hypcrkxt classifier t1.761 whcrc: the
recognition of a hypertcxt documcnt (e.g., R web page)
can be dramatically improved by j.ter.atively incorporiWng
the category informatioi~of other docutnents [hat point to o r
are pointed by this doculncnt.
9.2 Concluding Remarks
Watanabe [‘164] wrote in the prefacp of the 1972 book he
ed ited ,cn ti tlcd Fro I it i u s ~ 7 Pnt
f l ern. Xccox 1 iikiu H , tlw t ” PatLcrii
recognition is il fast-moving and prolifcrating disciplino. It
is not easy krr forin a well-balanced aiid well-informcd
summary view of thc ncwest developmmts i n this field. It
is still harder to have a vision c ~ fits future progress.”
34
IEEE TRANSACTIONS ON P A T E R N ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 1, JANUARY 2000
[25] C. Carpincto and C;. Nomatlo, "A Littirc Conceptual Clustcring
System and tts Application to Browsing Kctrieoal," Muchiii?
Lenviiirzg, vol. 24,110, 2, pp. 95-122, 1996.
The authors wuuld likc to thank the anonvmoiis reviewers,
and M. k?lillo, "An Iterative l'ruriing
al,d Drs, Rnz Pical,d,Til, Ho, and M~~~~~
(
~ for [heir
~
~ [26]i G. Castcllano,
~
~ A dM. Fanclli,
~
Algorithm for Fccdforwaxd Neural Kctworks," lt,EC, 'WUK
Nwud
critical and constructive comments a n d suggestioiis. 'rhc
N c ~ u u ~V~Os~8,110.
,.
3, pp 519-531, 1997.
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Constant Wcighling Functions," Ativoirrw iir iVciml riub~n~rttion
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h " . s s i i i ~ Sjysterus, G. 'I'csnuro, 11,s. Tourctxky, nncl T.K. Lccn,
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Classification Frror, and Complexity of Clansificntion Algurithms
Trunk, "A I'rublcm of I)iiiicnsiouiilit).: A Siinplc t.xainplc,"
in Pallern Recognition," lEL'11 Troris. I'ffftcvnAtidy.si.~ortd Mncfririr
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[ ISXI K. 'liimcr imd J . Ghosli, ".4nalysis of Decision Botiiidarics in
Liiwarly C:onitriiied Ncural Classifiers," I'/~tler~R e c o p iliotr, vol. 29,
pp. 341-348, 1996.
., .
Robert P.W. Duin studied applied physics at
Delft University of Technology in the Notherlands. In 1978, he received the PhD degree for a
:..
thesis on the accuracy of statistical pattern
{t59]S, Vnithyanathan and B. Dom, "Modcl Selection in Unsripcrvised
1 carnitrg with Applications tu nocumcnt Cluskrmg," /'roc. Sixth
recognizers. In his research, he included various
h t ' l Cor$ M o c h i ~ ~Imrniiis,
c
p ~133-443,
.
June 1994.
aspects of the automatic interpretation of measurements, learning syslems, and classifiers.
[I601 M. van Urcukclcn, R.P.W. Dum, U.M.J. Tax, and J.E. den Mactog,
Between 1980 and 1990, he studied and devel"Hondwrittcn Digit Recognition by Combinccl Classifiers,"
oped hardware architectures and software conK!yl?wwtih, vol. 34, no. 4,pp, 381-386,1998.
figurations for interactive image analysis. At
[ I61] V.N. Vapnik, Esthnlioit OJ D I ~ E I I I ~ IUnsed
I I I C 011
~ S EiirpiriunI DnM,
present, he is an associate professor of the faculty of appiied sciences
l3crliri: Spriugcr-Vrrlag, 1982.
I1621 V.N. Vapnik, Stofislicol Lcorriing 7%mr!/. New York John Wiley & of Delft University of Technology. His present research interest is in the
design and evaluation of learning algorithms for pattern recognition
Suns, 199H.
applications. This includes, in particular, neural natwork classifiers,
[ 163) S. Wntanabe, Pltitruiz Kccopiitiu!!: Htrnwr n d M w h i c o l . New
supporl vector classifiers, and classifier combining strategies. Recontly,
York: Wiley, 1985.
he began studying the possibilities of relational methods for pattern
11641Frotificrs uf k t l e r n Rmyqiiiinii. S. Watilnabc, cd , New York:
recognition.
Acadcmic Press, 1972.
[I651 AI<,Webb, "Mriltidimeiisional Scaling by Iterative Mwjorization
Using Radial Basis Futictions," P n f h w K e c q r i i t i o t ! , vol. 28, 110. 5,
Jlanchang Mao received the PhD degree in
pp. 753-759, 2995.
computer science from Michigan Stab Univer[ 1661S.M. W o k s and C.A. Kulikowski, Contptcr Syslems thnt Lrnrri.
sity, East Lansing, in 1994. He is currentiy a
Califnrnia: Mu~ganKaLdmann Piiblislirrs, 1991.
research stalf member at the IBM Almaden
[ I 671 M. Whindliam and A. Cutlvr, "It>formniioiiRatios for Validating
Research Center in San Jose, California. His
Mixtnrc Analysis," J, An!.St,itistiunI Assoa., vol. 87, pp. 1,188-1.,192,
research interests include paltern recognition,
'1492.
machine loarning, neural networks, information
[ 168111. Wdpcrt, "Stacked Ccncralization," Ncirrd Nctruorks, vol. 5, pp.
retrieval, web mining, document image analysis,
241-259, 3992.
OCR and handwriting recognition, image proces{I 691 A.K.C. Wong and D.C.C. !Aimg, "DECA: A Discvctc-Valued Data
sing, and computer vision. He receivod the
Clustering Algorithm," IEEE h".
P ~ I / ~ wA It f d y i s nird i\.lndliftt,
honorable mentlon award from the Pattern Recognition Society in
r n q s ~ ~ l f c c , 1, pp 342.349, 1979.
1993, and the 1996 /FEE Transactions on Neural Networks
[I701 K. Woods, W.P. KepIr\hcycr Jr., and K. hwycr. "Combinntion of outstanding paper award. He ak.0 received two IBM Research
Multiplc Classifiers Using Local Accuracy Iktifnates," I
ciivision awards in 1996 and 1998. and an IBM outstanding
/'/Nerti h n l y s i s rwd Moulriric Itrtellixcrw, vol. 19, no. 4, I)
technical achievement award in 1997. Ho was a guest co-editor
Apr. 1997.
of a special issue of IEEE Transnctlons on Nuerai Networks. He
11711 @.n. Xic, C.A. Laszlo, and K.K. Wildd, "Vcctnr Quantixatinn
has served as an associate editor of !€€E Transacfiuns on Neural
'I'cdinique f u r Nonpnminetric Classificr Design," IECE Tmns. Notworks, and as an editorial board member for Pattern Analysis
Potkm A i t i ! y s i s niid M d i w I r i t d i i p c c , vd. 15, no. 12, pp, 1,324- and Applications. He also served as a program committee member
1,330, 3993.
for several international conferences. He is a senior member of tho
[I721 T.. Xu, A. Krzymk, iind C.Y. Suen, "Mctliods lor Coiiibinitig
IEEE.
Multiple Classificrs and Thcir hpplicaiiuns in Ihidrvritteir
Character l<eecognition,"IEEE 'h",
. S y s ~ c n i ~Mmr,
,
nrrd Cybrmctics,
vol. 22, pp. 4'18-4375, 1992.
[ 1731 G.T. Toussaint, "The Uso nf Cniilext in I'nttcrn Recognition,"
IWtcrii Rccqyniiic~t, vol, 10, 110.3, pp. 189-204, 1978.
[ I741 K. Mohiudciin and J. Mao, "Opticnl C t m x l c r Recognition," Wilry
Cncyclopcdin a j Elrctrkd o d Elrxc\roiric Cizgiwi:ririg. 1.G. Webstcr,
~ d .\JOL
, 15, pp. 226-23
11Wilcy and Sons, Inc., 1
[I751 IKM, Haralick, "llr
Making in Context,"
Trnm.
~ ~ n nrjniysisis
t t ~ n ~~ ~~i n ~c i ~ rtti(!iligcIluc,
i ~ !
WI. 5, tlD. 4, pp. 417418, Mar. 1983.
I1761.S. Clwkrabarti, B. nom, and P. Iriyk, "linhaiiccd I-lypcrtcxt
Categorization Using Hyperlinks," AC"M SIGMOi3, Scattic; 1998
~~
Anll K. Jain is a university distinguished
professor in the Department of Computer
Science and Engineering at Michigan Slate
University, His research interests include statistical pattern recognition, Markov random fields,
texture analysis, neural networks. document
image analysis. fingerprint matching and 3D
object recognition. He received best paper
awards in 1987 and 1991 and certificates for
outstanding contributions in 1976, 1979, 1992,
and 1997 from the Pattern Recognition Society. He also received the
1996 /EF€ Transactions on Neural NefworhsOutstanding Paper Award.
He was the editor-imchief of the E E E Transaction on Pattern Analysis
and Machine Intelligence (1990-1994). He is the co-author of Algorithms
for Clustering Data, has edited lhe book Rod-Time Object Measurement
and Classification. and co-editod the books, Aoa/ysis and fnterpretation
of Range Images, Markov Random Fields, Artilicbt Neural Neiworks
and Pattern Recognition, 3 D Object Recognition, and BIOMETRICS:
Personal ldentilication in Nefworked Socieiy. He received a Fulbright
research award in 1998. He is a fellow of the IEEE and tAPR.